complements to classic topics of circles geometry
DESCRIPTION
We approach several themes of classical geometry of the circle and complete them with some original results, showing that not everything in traditional math is revealed, and that it still has an open character. The topics were chosen according to authors aspiration and attraction, as a poet writes lyrics about spring according to his emotions.TRANSCRIPT
Ion Patrascu | Florentin Smarandache
Complements to Classic Topics
of Circles Geometry
Pons Editions
Brussels | 2016
Complements to Classic Topics of Circles Geometry
1
Ion Patrascu | Florentin Smarandache
Complements to Classic Topics
of Circles Geometry
Ion Patrascu, Florentin Smarandache
2
In the memory of the first author's father
Mihail Patrascu and the second author's
mother Maria (Marioara) Smarandache,
recently passed to eternity...
Complements to Classic Topics of Circles Geometry
3
Ion Patrascu | Florentin Smarandache
Complements to Classic Topics
of Circles Geometry
Pons Editions
Brussels | 2016
Ion Patrascu, Florentin Smarandache
4
© 2016 Ion Patrascu & Florentin Smarandache
All rights reserved. This book is protected by
copyright. No part of this book may be
reproduced in any form or by any means,
including photocopying or using any
information storage and retrieval system
without written permission from the
copyright owners.
ISBN 978-1-59973-465-1
Complements to Classic Topics of Circles Geometry
5
Contents
Introduction ....................................................... 15
Lemoineâs Circles ............................................... 17
1st Theorem. ........................................................... 17
Proof. ................................................................. 17
2nd Theorem. ......................................................... 19
Proof. ................................................................ 19
Remark. ............................................................ 21
References. ........................................................... 22
Lemoineâs Circles Radius Calculus ..................... 23
1st Theorem ........................................................... 23
Proof. ................................................................ 23
Consequences. ................................................... 25
1st Proposition. ...................................................... 25
Proof. ................................................................ 25
2nd Proposition. .................................................... 26
Proof. ................................................................ 27
Remarks. ........................................................... 28
3rd Proposition. ..................................................... 28
Proof. ................................................................ 28
Remark. ............................................................ 29
References. ........................................................... 29
Radical Axis of Lemoineâs Circles ........................ 31
1st Theorem. ........................................................... 31
2nd Theorem. .......................................................... 31
1st Remark. ......................................................... 31
1st Proposition. ...................................................... 32
Proof. ................................................................ 32
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Comment. .......................................................... 34
2nd Remark. ....................................................... 34
2nd Proposition. .................................................... 34
References. ........................................................... 34
Generating Lemoineâs circles ............................. 35
1st Definition. ........................................................ 35
1st Proposition. ...................................................... 35
2nd Definition. ....................................................... 35
1st Theorem. .......................................................... 36
3rd Definition. ....................................................... 36
1st Lemma. ............................................................ 36
Proof. ................................................................ 36
Remark. ............................................................ 38
2nd Theorem. ......................................................... 38
Proof. ................................................................ 38
Further Remarks. .............................................. 40
References. ........................................................... 40
The Radical Circle of Ex-Inscribed Circles of a
Triangle ............................................................. 41
1st Theorem. .......................................................... 41
Proof. ................................................................ 41
2nd Theorem. ......................................................... 43
Proof. ................................................................ 43
Remark. ............................................................ 44
3rd Theorem. ......................................................... 44
Proof. ................................................................ 45
References. ........................................................... 48
The Polars of a Radical Center ........................... 49
1st Theorem. .......................................................... 49
Proof. ................................................................ 50
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2nd Theorem. .......................................................... 51
Proof. ................................................................ 52
Remarks. ........................................................... 52
References. ........................................................... 53
Regarding the First Droz-Farnyâs Circle ............. 55
1st Theorem. .......................................................... 55
Proof. ................................................................ 55
Remarks. ........................................................... 57
2nd Theorem. ......................................................... 57
Remark. ............................................................ 58
Definition. ............................................................ 58
Remark. ............................................................ 58
3rd Theorem. ......................................................... 58
Proof. ................................................................ 59
Remark. ............................................................ 60
4th Theorem. ......................................................... 60
Proof. ................................................................ 60
Reciprocally. ..................................................... 61
Remark. ............................................................ 62
References. ........................................................... 62
Regarding the Second Droz-Farnyâs Circle.......... 63
1st Theorem. .......................................................... 63
Proof. ................................................................ 63
1st Proposition. ...................................................... 65
Proof. ................................................................ 65
Remarks. ........................................................... 66
2nd Theorem. ......................................................... 66
Proof. ................................................................ 67
Reciprocally. ..................................................... 67
Remarks. ........................................................... 68
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2nd Proposition. .................................................... 68
Proof. ................................................................ 69
References. ........................................................... 70
Neubergâs Orthogonal Circles ............................. 71
1st Definition. ......................................................... 71
2nd Definition. ....................................................... 72
3rd Definition. ....................................................... 72
1st Proposition. ...................................................... 72
Proof. ................................................................ 73
Consequence. .................................................... 74
2nd Proposition. .................................................... 74
Proof. ................................................................ 74
4th Definition. ....................................................... 75
3rd Proposition. ..................................................... 75
Proof. ................................................................ 75
Reciprocally. ..................................................... 76
4th Proposition. ..................................................... 76
Proof. ................................................................ 77
References. ........................................................... 78
Lucasâs Inner Circles .......................................... 79
1. Definition of the Lucasâs Inner Circles .......... 79
Definition. ............................................................ 80
2. Calculation of the Radius of the A-Lucas Inner
Circle .................................................................... 80
Note. ................................................................. 81
1st Remark. ........................................................ 81
3. Properties of the Lucasâs Inner Circles .......... 81
1st Theorem. .......................................................... 81
Proof. ................................................................ 81
2nd Definition. ....................................................... 83
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Remark. ............................................................ 83
2nd Theorem. ......................................................... 84
3rd Definition. ....................................................... 84
3rd Theorem. ......................................................... 84
Proof. ................................................................ 85
Remarks. ........................................................... 86
4th Definition. ....................................................... 87
1st Property. .......................................................... 87
Proof. ................................................................ 87
2nd Property. ......................................................... 88
Proof. ................................................................ 88
References. ........................................................... 88
Theorems with Parallels Taken through a
Triangleâs Vertices and Constructions Performed
only with the Ruler ............................................ 89
1st Problem. ........................................................... 89
2nd Problem........................................................... 89
3rd Problem. .......................................................... 90
1st Lemma. ............................................................ 90
Proof. ................................................................ 91
Remark. ............................................................ 92
2nd Lemma. ........................................................... 92
Proof. ................................................................ 92
3rd Lemma. ........................................................... 93
Solution. ............................................................ 93
Constructionâs Proof. ........................................ 93
Remark. ............................................................ 94
1st Theorem. ......................................................... 95
Proof. ................................................................ 96
2nd Theorem. ....................................................... 97
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Proof. ................................................................ 98
Remark. ............................................................ 99
3rd Theorem. ......................................................... 99
Proof. ................................................................ 99
Reciprocally. ................................................... 100
Solution to the 1st problem. ................................. 101
Solution to the 2nd problem. ................................ 101
Solution to the 3rd problem. ............................... 102
References. ......................................................... 102
Apolloniusâs Circles of kth Rank ......................... 103
1st Definition. ...................................................... 103
2nd Definition. ..................................................... 103
1st Theorem. ........................................................ 103
Proof. .............................................................. 104
3rd Definition. ..................................................... 105
2nd Theorem. ....................................................... 105
Proof. .............................................................. 106
3rd Theorem. ....................................................... 107
Proof. .............................................................. 107
4th Theorem. ....................................................... 108
Proof. .............................................................. 108
1st Remark. ...................................................... 108
5th Theorem. ....................................................... 108
Proof. .............................................................. 109
6th Theorem. ....................................................... 109
Proof. .............................................................. 109
4th Definition. ...................................................... 110
1st Property. ......................................................... 110
2nd Remark. ...................................................... 110
7th Theorem. ........................................................ 111
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Proof. ............................................................... 111
References. .......................................................... 112
Apolloniusâs Circle of Second Rank ................... 113
1st Definition. ....................................................... 113
1st Theorem. ......................................................... 113
1st Proposition. ..................................................... 114
Proof. ............................................................... 114
Remarks. .......................................................... 115
2nd Definition. ...................................................... 115
3rd Definition. ...................................................... 116
2nd Proposition. ................................................... 116
Proof. ............................................................... 116
Remarks. .......................................................... 118
Open Problem. ..................................................... 119
References. ......................................................... 120
A Sufficient Condition for the Circle of the 6 Points
to Become Eulerâs Circle ................................... 121
1st Definition. ....................................................... 121
1st Remark. ....................................................... 122
2nd Definition. ...................................................... 122
2nd Remark. ...................................................... 122
1st Proposition. ..................................................... 122
Proof. ............................................................... 122
3rd Remark. ...................................................... 123
3rd Definition. ...................................................... 123
4th Remark. ..................................................... 124
1st Theorem. ........................................................ 124
Proof. .............................................................. 124
4th Definition. ..................................................... 126
2nd Proposition. .................................................. 126
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1st Proof. .......................................................... 126
2nd Proof. .......................................................... 127
5th Remark. ..................................................... 128
References. ......................................................... 128
An Extension of a Problem of Fixed Point ......... 129
Proof. .............................................................. 130
1st Remark. ....................................................... 132
2nd Remark. ...................................................... 132
3rd Remark. ...................................................... 133
References. .......................................................... 133
Some Properties of the Harmonic Quadrilateral 135
1st Definition. ....................................................... 135
2nd Definition. ...................................................... 135
1st Proposition. ..................................................... 135
2nd Proposition. .................................................. 136
Proof. .............................................................. 136
Remark 1. ......................................................... 137
3rd Proposition. .................................................... 137
Remark 2. ........................................................ 138
Proposition 4. ..................................................... 138
Proof. .............................................................. 139
3rd Remark. ..................................................... 140
5th Proposition. ................................................... 140
6th Proposition. ................................................... 140
Proof. .............................................................. 140
3rd Definition. ..................................................... 142
4th Remark. ..................................................... 142
7th Proposition. ................................................... 143
Proof. .............................................................. 143
5th Remark. ..................................................... 144
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8th Proposition. ................................................... 145
Proof. .............................................................. 145
4th Definition. ..................................................... 146
6th Remark. ..................................................... 146
9th Proposition. ................................................... 146
Proof. .............................................................. 146
10th Proposition. ..................................................147
Proof. ...............................................................147
7th Remark....................................................... 148
11th Proposition. .................................................. 148
Proof. .............................................................. 149
References. ......................................................... 149
Triangulation of a Triangle with Triangles having
Equal Inscribed Circles ..................................... 151
Solution. .............................................................. 151
Building the point D. ........................................... 153
Proof. .............................................................. 154
Discussion. ....................................................... 155
Remark. .......................................................... 156
Open problem. .................................................... 156
An Application of a Theorem of Orthology ........ 157
Proposition. ......................................................... 157
Proof. .............................................................. 158
Remark. .......................................................... 160
Note. ............................................................... 160
References. ......................................................... 162
The Dual of a Theorem Relative to the Orthocenter
of a Triangle ..................................................... 163
1st Theorem. ........................................................ 163
Proof. .............................................................. 164
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Note. ............................................................... 165
2nd Theorem. ....................................................... 165
Proof. .............................................................. 166
References. ......................................................... 168
The Dual Theorem Concerning Aubertâs Line .... 169
1st Theorem. ........................................................ 169
Proof. .............................................................. 169
Note. ................................................................ 171
1st Definition. ....................................................... 171
Note. ................................................................ 171
2nd Definition. ...................................................... 172
2nd Theorem. ........................................................ 172
Note. ................................................................ 172
3rd Theorem. ........................................................ 172
Proof. ............................................................... 173
Notes. ...............................................................174
4th Theorem. ........................................................174
Proof. ...............................................................174
References. ..........................................................176
Bibliography ..................................................... 177
Complements to Classic Topics of Circles Geometry
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Introductory Note
We approach several themes of classical
geometry of the circle and complete them
with some original results, showing that not
everything in traditional math is revealed,
and that it still has an open character.
The topics were chosen according to
authorsâ aspiration and attraction, as a poet
writes lyrics about spring according to his
emotions.
Ion Patrascu, Florentin Smarandache
16
Complements to Classic Topics of Circles Geometry
17
Lemoineâs Circles
In this article, we get to Lemoine's circles
in a different manner than the known one.
1st Theorem.
Let đŽđ”đ¶ a triangle and đŸ its simedian center. We
take through K the parallel đŽ1đŽ2 to đ”đ¶, đŽ1 â (đŽđ”), đŽ2 â
(đŽđ¶); through đŽ2 we take the antiparallels đŽ2đ”1 to đŽđ”
in relation to đ¶đŽ and đ¶đ” , đ”1 â (đ”đ¶) ; through đ”1 we
take the parallel đ”1đ”2 to đŽđ¶ , đ”2 â đŽđ”; through đ”2 we
take the antiparallels đ”1đ¶1 to đ”đ¶ , đ¶1 â (đŽđ¶) , and
through đ¶1 we take the parallel đ¶1đ¶2 to đŽđ”, đ¶1 â (đ”đ¶).
Then:
i. đ¶2đŽ1 is an antiparallel of đŽđ¶;
ii. đ”1đ”2 â© đ¶1đ¶2 = {đŸ};
iii. The points đŽ1, đŽ2 , đ”1 , đ”2, đ¶1, đ¶2 are
concyclical (the first Lemoineâs circle).
Proof.
i. The quadrilateral đ”đ¶2đŸđŽ is a
parallelogram, and its center, i.e. the
middle of the segment (đ¶2đŽ1), belongs to
the simedian đ”đŸ; it follows that đ¶2đŽ2 is
an antiparallel to đŽđ¶(see Figure 1).
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ii. Let {đŸâČ} = đŽ1đŽ2 â© đ”1đ”2 , because the
quadrilateral đŸâČđ”1đ¶đŽ2 is a parallelogram;
it follows that đ¶đŸâČ is a simedian; also,
đ¶đŸ is a simedian, and since đŸ,đŸâČ â đŽ1đŽ2, it
follows that we have đŸâČ = đŸ.
iii. đ”2đ¶1 being an antiparallel to đ”đ¶ and
đŽ1đŽ2 â„ đ”đ¶ , it means that đ”2đ¶1 is an
antiparallel to đŽ1đŽ2 , so the points
đ”2, đ¶1, đŽ2, đŽ1 are concyclical. From đ”1đ”2 â„
đŽđ¶ , âąđ”2đ¶1đŽ ⥠âąđŽđ”đ¶ , âąđ”1đŽ2đ¶ ⥠âąđŽđ”đ¶ we
get that the quadrilateral đ”2đ¶1đŽ2đ”1 is an
isosceles trapezoid, so the points
đ”2, đ¶1, đŽ2, đ”1 are concyclical. Analogously,
it can be shown that the quadrilateral
đ¶2đ”1đŽ2đŽ1 is an isosceles trapezoid,
therefore the points đ¶2, đ”1, đŽ2, đŽ1 are
concyclical.
Figure 1
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From the previous three quartets of concyclical
points, it results the concyclicity of the points
belonging to the first Lemoineâs circle.
2nd Theorem.
In the scalene triangle đŽđ”đ¶, let K be the simedian
center. We take from đŸ the antiparallel đŽ1đŽ2 to đ”đ¶ ;
đŽ1 â đŽđ”, đŽ2 â đŽđ¶; through đŽ2 we build đŽ2đ”1 â„ đŽđ”; đ”1 â
(đ”đ¶), then through đ”1 we build đ”1đ”2 the antiparallel
to đŽđ¶ , đ”2 â (đŽđ”), and through đ”2 we build đ”2đ¶1 â„ đ”đ¶ ,
đ¶1 â đŽđ¶ , and, finally, through đ¶1 we take the
antiparallel đ¶1đ¶2 to đŽđ”, đ¶2 â (đ”đ¶).
Then:
i. đ¶2đŽ1 â„ đŽđ¶;
ii. đ”1đ”2 â© đ¶1đ¶2 = {đŸ};
iii. The points đŽ1, đŽ2, đ”1, đ”2, đ¶1, đ¶2 are
concyclical (the second Lemoineâs circle).
Proof.
i. Let {đŸâČ} = đŽ1đŽ2 â© đ”1đ”2 , having âąđŽđŽ1đŽ2 =
âąđŽđ¶đ” and âąđ”đ”1đ”2 ⥠âąđ”đŽđ¶ because đŽ1đŽ2
Èi đ”1đ”2 are antiparallels to đ”đ¶ , đŽđ¶ ,
respectively, it follows that âąđŸâČđŽ1đ”2 âĄ
âąđŸâČđ”2đŽ1 , so đŸâČđŽ1 = đŸâČđ”2 ; having đŽ1đ”2 â„
đ”1đŽ2 as well, it follows that also đŸâČđŽ2 =
đŸâČđ”1 , so đŽ1đŽ2 = đ”1đ”2 . Because đ¶1đ¶2 and
đ”1đ”2 are antiparallels to đŽđ” and đŽđ¶ , we
Ion Patrascu, Florentin Smarandache
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have đŸ"đ¶2 = đŸ"đ”1; we noted {đŸ"} = đ”1đ”2 â©
đ¶1đ¶2; since đ¶1đ”2 â„ đ”1đ¶2, we have that the
triangle đŸ"đ¶1đ”2 is also isosceles, therefore
đŸ"đ¶1 = đ¶1đ”2, and we get that đ”1đ”2 = đ¶1đ¶2.
Let {đŸâČâČâČ} = đŽ1đŽ2 â© đ¶1đ¶2 ; since đŽ1đŽ2 and
đ¶1đ¶2 are antiparallels to đ”đ¶ and đŽđ” , we
get that the triangle đŸâČâČâČđŽ2đ¶1 is isosceles,
so đŸâČâČâČđŽ2 = đŸâČâČâČđ¶1 , but đŽ1đŽ2 = đ¶1đ¶2 implies
that đŸâČâČâČđ¶2 = đŸâČâČâČđŽ1 , then âąđŸâČâČâČđŽ1đ¶2 âĄ
âąđŸâČâČâČđŽ2đ¶1 and, accordingly, đ¶2đŽ1 â„ đŽđ¶.
Figure 2
ii. We noted {đŸâČ} = đŽ1đŽ2 â© đ”1đ”2 ; let {đ} =
đ”2đ¶1 â© đ”1đŽ2 ; obviously, đ”đ”1đđ”2 is a
parallelogram; if đŸ0 is the middle of
(đ”1đ”2), then đ”đŸ0 is a simedian, since đ”1đ”2
is an antiparallel to đŽđ¶, and the middle of
the antiparallels of đŽđ¶ are situated on the
Complements to Classic Topics of Circles Geometry
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simedian đ”đŸ . If đŸ0 â K, then đŸ0đŸ â„ đŽ1đ”2
(because đŽ1đŽ2 = đ”1đ”2 and đ”1đŽ2 â„ đŽ1đ”2 ),
on the other hand, đ”, đŸ0, K are collinear
(they belong to the simedian đ”đŸ ),
therefore đŸ0đŸ intersects đŽđ” in đ”, which is
absurd, so đŸ0 =K, and, accordingly, đ”1đ”2 â©
đŽ1đŽ2 = {đŸ} . Analogously, we prove that
đ¶1đ¶2 â© đŽ1đŽ2 = {đŸ}, so đ”1đ”2 â© đ¶1đ¶2 = {đŸ}.
iii. K is the middle of the congruent
antiparalells đŽ1đŽ2 , đ”1đ”2 , đ¶1đ¶2 , so đŸđŽ1 =
đŸđŽ2 = đŸđ”1 = đŸđ”2 = đŸđ¶1 = đŸđ¶2 . The
simedian center đŸ is the center of the
second Lemoineâs circle.
Remark.
The center of the first Lemoineâs circle is the
middle of the segment [đđŸ], where đ is the center of
the circle circumscribed to the triangle đŽđ”đ¶. Indeed,
the perpendiculars taken from đŽ, đ”, đ¶ on the
antiparallels đ”2đ¶1 , đŽ1đ¶2 , đ”1đŽ2 respectively pass
through O, the center of the circumscribed circle (the
antiparallels have the directions of the tangents taken
to the circumscribed circle in đŽ, đ”, đ¶). The mediatrix of
the segment đ”2đ¶1 pass though the middle of đ”2đ¶1 ,
which coincides with the middle of đŽđŸ , so is the
middle line in the triangle đŽđŸđ passing through the
middle of (đđŸ) . Analogously, it follows that the
mediatrix of đŽ1đ¶2 pass through the middle đż1 of [đđŸ].
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References.
[1] D. Efremov: Noua geometrie a triunghiului [The New Geometry of the Triangle], translation from Russian into Romanian by Mihai MiculiÈa,
Cril Publishing House, Zalau, 2010. [2] Gh. Mihalescu: Geometria elementelor remarcabile
[The Geometry of the Outstanding Elements],
Tehnica Publishing House, Bucharest, 1957. [3] Ion Patrascu, Gh. Margineanu: Cercul lui Tucker
[Tuckerâs Circle], in Alpha journal, year XV, no. 2/2009.
Complements to Classic Topics of Circles Geometry
23
Lemoineâs Circles Radius
Calculus
For the calculus of the first Lemoineâs
circle, we will first prove:
1st Theorem (E. Lemoine â 1873)
The first Lemoineâs circle divides the sides of a
triangle in segments proportional to the squares of the
triangleâs sides.
Each extreme segment is proportional to the
corresponding adjacent side, and the chord-segment
in the Lemoineâs circle is proportional to the square of
the side that contains it.
Proof.
We will prove that đ”đ¶2
đ2 = đ¶2đ”1
đ2 = đ”1đ¶
đ2 .
In figure 1, đŸ represents the symmedian center
of the triangle đŽđ”đ¶ , and đŽ1đŽ2 ; đ”1đ”2 ; đ¶1đ¶2 represent
Lemoine parallels.
The triangles đ”đ¶2đŽ1 ; đ¶đ”1đŽ2 and đŸđ¶2đŽ1 have
heights relative to the sides đ”đ¶2; đ”1đ¶ and đ¶2đ”1 equal
(đŽ1đŽ2 â„ đ”đ¶).
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Hence: đŽđđđâ đ”đŽ1đ¶2
đ”đ¶2 =
đŽđđđâ đŸđ¶2đŽ1
đ¶2đ”1 =
đŽđđđâ đ¶đ”1đŽ2
đ”1đ¶ . (1)
Figure 1
On the other hand: đŽ1đ¶2 and đ”1đŽ2 being
antiparallels with respect to đŽđ¶ and đŽđ”, it follows that
Îđ”đ¶2đŽ1~đ„đ”đŽđ¶ and Îđ¶đ”1đŽ2~đ„đ¶đŽđ” , likewise đŸđ¶2 â„ đŽđ¶
implies: ÎđŸđ¶2đ”1~đ„đŽđ”đ¶.
We obtain: đŽđđđâ đ”đ¶2đŽ1
đŽđđđâ đŽđ”đ¶ =
đ”đ¶22
đ2 ;
đŽđđđâ đŸđ¶2đ”1
đŽđđđâ đŽđ”đ¶=
đ¶2đ”12
đ2 ;
đŽđđđâ đ¶đ”1đŽ2
đŽđđđâ đŽđ”đ¶ =
đ¶đ”12
đ2 . (2)
If we denote đŽđđđâ đŽđ”đ¶ = đ, we obtain from the
relations (1) and (2) that: đ”đ¶2
đ2 = đ¶2đ”1
đ2 = đ”1đ¶
đ2 .
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25
Consequences.
1. According to the 1st Theorem, we find that:
đ”đ¶2 =đđ2
đ2+đ2+đ2; đ”1đ¶ =đđ2
đ2+đ2+đ2; đ”1đ¶2 =đ3
đ2+đ2+đ2 .
2. We also find that: đ”1đ¶2
đ3 = đŽ2đ¶1
đ3 = đŽ1đ”2
đ3 ,
meaning that:
âThe chords determined by the first Lemoineâs
circle on the triangleâs sides are proportional to
the cubes of the sides.â
Due to this property, the first Lemoineâs circle is
known in England by the name of triplicate ratio circle.
1st Proposition.
The radius of the first Lemoineâs circle, đ đż1 is
given by the formula:
đ đż1
2 =1
4âđ 2(đ2+đ2+đ2)+ đ2đ2đ2
(đ2+đ2+đ2)2, (3)
where đ represents the radius of the circle inscribed
in the triangle đŽđ”đ¶.
Proof.
Let đż be the center of the first Lemoineâs circle
that is known to represent the middle of the segment
(đđŸ) â đ being the center of the circle inscribed in the
triangle đŽđ”đ¶.
Ion Patrascu, Florentin Smarandache
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Considering C1, we obtain đ”đ”1 =đ (đ2+đ2)
đ2+đ2+đ2 .
Taking into account the power of point đ” in
relation to the first Lemoineâs circle, we have:
đ”đ¶2 â đ”đ”1 = đ”đ2 â đżđ2,
(đ”đ is the tangent traced from đ” to the first Lemoineâs
circle, see Figure 1).
Hence: đ đż1
2 = đ”đż2 â đ”đ¶2 â đ”đ”1. (4)
The median theorem in triangle đ”đđŸ implies
that:
đ”đż2 =2â(đ”đŸ2+đ”đ2)âđđŸ2
4 .
It is known that đŸ =(đ2+đ2)âđđ
đ2+đ2+đ2 ; đđ =2đđâđđ
đ2+đ2 ,
where đđ and đđ are the lengths of the symmedian
and the median from đ”, and đđŸ2 = đ 2 â 3đ2đ2đ2
(đ2+đ2+đ2) , see
(3).
Consequently: đ”đŸ2 =2đ2đ2(đ2+đ2)âđ2đ2đ2
(đ2+đ2+đ2)2 , and
4đ”đż2 = đ 2 +4đ2đ2(đ2+đ2)+đ2đ2đ2
(đ2+đ2+đ2)2 .
As: đ”đ¶2 â đ”đ”1 = đ2đ2(đ2+đ2)
(đ2+đ2+đ2)2 , by replacing in (4),
we obtain formula (3).
2nd Proposition.
The radius of the second Lemoineâs circle, đ đż2, is
given by the formula:
đ đż2=
đđđ
đ2+đ2+đ2 . (5)
Complements to Classic Topics of Circles Geometry
27
Proof.
Figure 2
In Figure 2 above, đŽ1đŽ2; đ”1đ”2; đ¶1đ¶2 are Lemoine
antiparallels traced through symmedian center đŸ that
is the center of the second Lemoineâs circle, thence:
đ đż2= đŸđŽ1 = đŸđŽ2.
If we note with đ and đ the feet of the
symmedian and the median from đŽ, it is known that:
đŽđŸ
đŸđ=
đ2+đ2
đ2 .
From the similarity of triangles đŽđŽ2đŽ1 and đŽđ”đ¶,
we have: đŽ1đŽ2
đ”đ¶=
đŽđŸ
đŽđ .
But: đŽđŸ
đŽđ=
đ2+đ2
đ2+đ2+đ2 and đŽđ =2đđ
đ2+đ2 â đđ.
đŽ1đŽ2 = 2đ đż2, đ”đ¶ = đ, therefore:
đ đż2=
đŽđŸâđ
2đđ ,
and as đŽđŸ = 2đđ â đđ
đ2+đ2+đ2 , formula (5) is a consequence.
Ion Patrascu, Florentin Smarandache
28
Remarks.
1. If we use đĄđđ =4đ
đ2+đ2+đ2, đ being the Brocardâs
angle (see [2]), we obtain: đ đż2= đ â đĄđđ.
2. If, in Figure 1, we denote with đ1 the middle
of the antiparallel đ”2đ¶1, which is equal to đ đż2 (due to
their similarity), we thus find from the rectangular
triangle đżđ1đ¶1 that:
đżđ¶12 = đżđ1
2 + đ1đ¶12 , but đżđ1
2 =1
4đ2 and đ1đ¶2 =
1
2đ đż2
; it follows that:
đ đż1
2 =1
4(đ 2 + đ đż2
2 ) =đ 2
4(1 + đĄđ2đ).
We obtain:
đ đż1=
đ
2â â1 + đĄđ2đ .
3rd Proposition.
The chords determined by the sides of the
triangle in the second Lemoineâs circle are
respectively proportional to the opposing angles
cosines.
Proof.
đŸđ¶2đ”1 is an isosceles triangle, âąđŸđ¶2đ”1 =
âąđŸđ”1đ¶2 = âąđŽ; as đŸđ¶2 = đ đż2 we have that cos đŽ =
đ¶2đ”1
2đ đż2
,
deci đ¶2đ”1
cosđŽ= 2đ đż2
, similary: đŽ2đ¶1
cosđ”=
đ”2đŽ1
cosđ¶= 2đ đż2.
Complements to Classic Topics of Circles Geometry
29
Remark.
Due to this property of the Lemoineâs second
circle, in England this circle is known as the cosine
circle.
References.
[1] D. Efremov, Noua geometrie a triunghiului [The
New Geometry of the Triangle], translation from Russian into Romanian by Mihai MiculiÈa,
Cril Publishing House, Zalau, 2010. [2] F. Smarandache and I. Patrascu, The Geometry of
Homological Triangles, The Education Publisher,
Ohio, USA, 2012. [3] I. Patrascu and F. Smarandache, Variance on
Topics of Plane Geometry, Educational
Publisher, Ohio, USA, 2013.
Ion Patrascu, Florentin Smarandache
30
Complements to Classic Topics of Circles Geometry
31
Radical Axis of Lemoineâs
Circles
In this article, we emphasize the radical
axis of the Lemoineâs circles.
For the start, let us remind:
1st Theorem.
The parallels taken through the simmedian
center đŸ of a triangle to the sides of the triangle
determine on them six concyclic points (the first
Lemoineâs circle).
2nd Theorem.
The antiparallels taken through the triangleâs
simmedian center to the sides of a triangle determine
six concyclic points (the second Lemoineâs circle).
1st Remark.
If đŽđ”đ¶ is a scalene triangle and đŸ is its
simmedian center, then đż , the center of the first
Lemoineâs circle, is the middle of the segment [đđŸ],
where đ is the center of the circumscribed circle, and
Ion Patrascu, Florentin Smarandache
32
the center of the second Lemoineâs circle is đŸ . It
follows that the radical axis of Lemoineâs circles is
perpendicular on the line of the centers đżđŸ, therefore
on the line đđŸ.
1st Proposition.
The radical axis of Lemoineâs circles is perpendi-
cular on the line đđŸ raised in the simmedian center đŸ.
Proof.
Let đŽ1đŽ2 be the antiparallel to đ”đ¶ taken through
đŸ, then đŸđŽ1 is the radius đ đż2 of the second Lemoineâs
circle; we have:
đ đż2=
đđđ
đ2+đ2+đ2 .
Figure 1
Complements to Classic Topics of Circles Geometry
33
Let đŽ1âČ đŽ2
âČ be the Lemoineâs parallel taken to đ”đ¶;
we evaluate the power of đŸ towards the first
Lemoineâs circle. We have:
đŸđŽ1âČ â đŸđŽ2
âČ = đżđŸ2 â đ đż1
2 . (1)
Let đ be the simmedian leg from đŽ; it follows
that: đŸđŽ1
âČ
đ”đ=
đŽđŸ
đŽđâ
đŸđŽ2âČ
đđ¶ .
We obtain:
đŸđŽ1âČ = đ”đ â
đŽđŸ
đŽđ and đŸđŽ2
âČ = đđ¶ âđŽđŸ
đŽđ ,
but đ”đ
đđ¶=
đ2
đ2 and đŽđŸ
đŽđ=
đ2+đ2
đ2+đ2+đ2 .
Therefore:
đŸđŽ1âČ â đŸđŽ2
âČ = âđ”đ â đđ¶ â (đŽđŸ
đŽđ)2
=âđ2đ2đ2
(đ2 + đ2)2;
(đ2+đ2)2
(đ2+đ2+đ2)2= âđ đż2
2 . (2)
We draw the perpendicular in đŸ on the line
đżđŸ and denote by đ and đ its intersection to the first
Lemoineâs circle.
We have đŸđ â đŸđ = âđ đż2
2 ; by the other hand,
đŸđ = đŸđ (đđ is a chord which is perpendicular to the
diameter passing through đŸ).
It follows that đŸđ = đŸđ = đ đż2, so đ and đ are
situated on the second Lemoineâs circle.
Because đđ is a chord which is common to the
Lemoineâs circles, it obviously follows that đđ is the
radical axis.
Ion Patrascu, Florentin Smarandache
34
Comment.
Equalizing (1) and (2), or by the Pythagorean
theorem in the triangle đđŸđż, we can calculate đ đż1.
It is known that: đđŸ2 = đ 2 â3đ2đ2đ2
(đ2+đ2+đ2)2 , and
since đżđŸ =1
2đđŸ, we find that:
đ đż1
2 =1
4â [đ 2 +
đ2đ2đ2
(đ2+đ2+đ2)2].
2nd Remark.
The 1st Proposition, ref. the radical axis of the
Lemoineâs circles, is a particular case of the following
Proposition, which we leave to the reader to prove.
2nd Proposition.
If đ(đ1, đ 1) Ći đ(đ2, đ 2) are two circles such as
the power of center đ1 towards đ(đ2, đ 2) is âđ 12, then
the radical axis of the circles is the perpendicular in
đ1 on the line of centers đ1đ2.
References.
[1] F. Smarandache, Ion Patrascu: The Geometry of Homological Triangles, Education Publisher, Ohio, USA, 2012.
[2] Ion Patrascu, F. Smarandache: Variance on Topics of Plane Geometry, Education Publisher, Ohio, USA, 2013.
Complements to Classic Topics of Circles Geometry
35
Generating Lemoineâs circles
In this paper, we generalize the theorem
relative to the first Lemoineâs circle and
thereby highlight a method to build
Lemoineâs circles.
Firstly, we review some notions and results.
1st Definition.
It is called a simedian of a triangle the symmetric
of a median of the triangle with respect to the internal
bisector of the triangle that has in common with the
median the peak of the triangle.
1st Proposition.
In the triangle đŽđ”đ¶, the cevian đŽđ, đ â (đ”đ¶), is a
simedian if and only if đđ”
đđ¶= (
đŽđ”
đŽđ¶)2. For Proof, see [2].
2nd Definition.
It is called a simedian center of a triangle (or
Lemoineâs point) the intersection of triangleâs
simedians.
Ion Patrascu, Florentin Smarandache
36
1st Theorem.
The parallels to the sides of a triangle taken
through the simedian center intersect the triangleâs
sides in six concyclic points (the first Lemoineâs circle
- 1873).
A Proof of this theorem can be found in [2].
3rd Definition.
We assert that in a scalene triangle đŽđ”đ¶ the line
đđ, where đ â đŽđ” and đ â đŽđ¶, is an antiparallel to đ”đ¶
if âąđđđŽ ⥠âąđŽđ”đ¶.
1st Lemma.
In the triangle đŽđ”đ¶ , let đŽđ be a simedian, đ â
(đ”đ¶). If đ is the middle of the segment (đđ), having
đ â (đŽđ”) and đ â (đŽđ¶), belonging to the simedian đŽđ,
then đđ and đ”đ¶ are antiparallels.
Proof.
We draw through đ and đ, đđ â„ đŽđ¶ and đđ â„ đŽđ”,
đ , đ â (đ”đ¶) , see Figure 1. Let {đ} = đđ â© đđ ; since
đđ = đđ and đŽđđđ is a parallelogram, it follows that
đ â đŽđ.
Complements to Classic Topics of Circles Geometry
37
Figure 1.
Thales's Theorem provides the relations: đŽđ
đŽđ¶=
đ”đ
đ”đ¶; (1)
đŽđ”
đŽđ=
đ”đ¶
đ¶đ . (2)
From (1) and (2), by multiplication, we obtain: đŽđ
đŽđâđŽđ”
đŽđ¶=
đ”đ
đđ¶ . (3)
Using again Thales's Theorem, we obtain: đ”đ
đ”đ=
đŽđ
đŽđ , (4)
đđ¶
đđ¶=
đŽđ
đŽđ . (5)
From these relations, we get đ”đ
đ”đ=
đđ¶
đđ¶ , (6)
or đ”đ
đđ¶=
đ”đ
đđ¶ . (7)
In view of Proposition 1, the relations (7) and (3)
lead to đŽđ
đŽđ”=
đŽđ”
đŽđ¶, which shows that âđŽđđ~âđŽđ¶đ” , so
âąđŽđđ ⥠âąđŽđ”đ¶.
Ion Patrascu, Florentin Smarandache
38
Therefore, đđ and đ”đ¶ are antiparallels in
relation to đŽđ” and đŽđ¶.
Remark.
1. The reciprocal of Lemma 1 is also valid,
meaning that if đ is the middle of the antiparallel đđ
to đ”đ¶, then đ belongs to the simedian from đŽ.
2nd Theorem. (Generalization of the 1st Theorem)
Let đŽđ”đ¶ be a scalene triangle and đŸ its simedian
center. We take đ â đŽđŸ and draw đđ â„ đŽđ”,đđ â„ đŽđ¶ ,
where đ â đ”đŸ, đ â đ¶đŸ. Then:
i. đđ â„ đ”đ¶;
ii. đđ,đđ and đđ intersect the sides of
triangle đŽđ”đ¶ in six concyclic points.
Proof.
In triangle đŽđ”đ¶, let đŽđŽ1, đ”đ”1, đ¶đ¶1 the simedians
concurrent in đŸ (see Figure 2).
We have from Thales' Theorem that: đŽđ
đđŸ=
đ”đ
đđŸ; (1)
đŽđ
đđŸ=
đ¶đ
đđŸ . (2)
From relations (1) and (2), it follows that đ”đ
đđŸ=
đ¶đ
đđŸ, (3)
which shows that đđ â„ đ”đ¶.
Complements to Classic Topics of Circles Geometry
39
Let đ , đ, đ,đ, đ, đ be the intersection points of
the parallels đđ,đđ,đđ of the sides of the triangles to
the other sides.
Obviously, by construction, the quadrilaterals
đŽđđđ; đ¶đđđ; đ”đ đđ are parallelograms.
The middle of the diagonal đđ falls on đŽđ, so on
the simedian đŽđŸ, and from 1st Lemma we get that đđ
is an antiparallel to đ”đ¶.
Since đđ â„ đ”đ¶ , it follows that đđ and đđ are
antiparallels, therefore the points đ, đ, đ, đ are
concyclic (4).
Figure 2.
Analogously, we show that the points đ, đ, đ , đ
are concyclic (5). From đđ and đ”đ¶ antiparallels, đđ
and đŽđ” antiparallels, we have that âąđđđŽ ⥠âąđŽđ”đ¶ and
âąđđđ¶ ⥠âąđŽđ”đ¶ , therefore: âąđđđŽ ⥠âąđđđ¶ , and since
Ion Patrascu, Florentin Smarandache
40
đđ â„ đŽđ¶, it follows that the trapeze đđđđ is isosceles,
therefore the points đ, đ, đ, đ are concyclic (6).
The relations (4), (5), (6) drive to the
concyclicality of the points đ , đ, đ, đ,đ, đ, and the
theorem is proved.
Further Remarks.
2. For any point đ found on the simedian đŽđŽ1, by
performing the constructions from hypothesis, we get
a circumscribed circle of the 6 points of intersection
of the parallels taken to the sides of triangle.
3. The 2nd Theorem generalizes the 1st Theorem
because we get the second in the case the parallels are
taken to the sides through the simedian center đ.
4. We get a circle built as in 2nd Theorem from
the first Lemoineâs circle by homothety of pole đ and
of ratio đ â â.
5. The centers of Lemoineâs circles built as above
belong to the line đđŸ , where đ is the center of the
circle circumscribed to the triangle đŽđ”đ¶.
References.
[1] Exercices de Géométrie, par F.G.M., HuitiÚme
édition, Paris VIe, Librairie Générale, 77, Rue Le Vaugirard.
[2] Ion Patrascu, Florentin Smarandache: Variance on
topics of Plane Geometry, Educational Publishing, Columbus, Ohio, 2013.
Complements to Classic Topics of Circles Geometry
41
The Radical Circle of Ex-
Inscribed Circles of a Triangle
In this article, we prove several theorems
about the radical center and the radical
circle of ex-inscribed circles of a triangle
and calculate the radius of the circle from
vectorial considerations.
1st Theorem.
The radical center of the ex-inscribed circles of
the triangle đŽđ”đ¶ is the Spieckerâs point of the triangle
(the center of the circle inscribed in the median
triangle of the triangle đŽđ”đ¶).
Proof.
We refer in the following to the notation in
Figure 1. Let đŒđ , đŒđ , đŒđ be the centers of the ex-inscribed
circles of a triangle (the intersections of two external
bisectors with the internal bisector of the other angle).
Using tangents property taken from a point to a circle
to be congruent, we calculate and find that:
đŽđčđ = đŽđžđ = đ”đ·đ = đ”đčđ = đ¶đ·đ = đ¶đžđ = đ,
đ”đ·đ = đ”đčđ = đ¶đ·đ = đ¶đžđ = đ â đ,
Ion Patrascu, Florentin Smarandache
42
đ¶đžđ = đ¶đ·đ = đŽđčđ = đŽđžđ = đ â đ,
đŽđčđ = đŽđžđ = đ”đčđ = đ”đ·đ = đ â đ.
If đŽ1 is the middle of segment đ·đđ·đ , it follows
that đŽ1 has equal powers to the ex-inscribed circles
(đŒđ) and (đŒđ). Of the previously set, we obtain that đŽ1
is the middle of the side đ”đ¶.
Figure 1.
Also, the middles of the segments đžđđžđ and đčđđčđ,
which we denote đ and đ, have equal powers to the
circles (đŒđ) and (đŒđ).
The radical axis of the circles ( đŒđ ), ( đŒđ ) will
include the points đŽ1, đ, đ.
Because đŽđžđ = đŽđčđ and đŽđžđ = đŽđčđ, it follows that
đŽđ = đŽđ and we find that âąđŽđđ =1
2âąđŽ, therefore the
Complements to Classic Topics of Circles Geometry
43
radical axis of the ex-inscribed circles (đčđ) and (đŒđ) is
the parallel taken through the middle đŽ1 of the side đ”đ¶
to the bisector of the angle đ”đŽđ¶.
Denoting đ”1 and đ¶1 the middles of the sides đŽđ¶,
đŽđ”, respectively, we find that the radical center of the
ex-inscribed circles is the center of the circle inscribed
in the median triangle đŽ1đ”1đ¶1 of the triangle đŽđ”đ¶.
This point, denoted đ, is the Spieckerâs point of
the triangle ABC.
2nd Theorem.
The radical center of the inscribed circle (đŒ) and
of the đ” âex-inscribed and đ¶ âex-inscribed circles of
the triangle đŽđ”đ¶ is the center of the đŽ1 â ex-inscribed
circle of the median triangle đŽ1đ”1đ¶1, corresponding to
the triangle đŽđ”đ¶).
Proof.
If đž is the contact of the inscribed circle with đŽđ¶
and đžđ is the contact of the đ” âex-inscribed circle with
đŽđ¶ , it is known that these points are isotomic,
therefore the middle of the segment đžđžđ is the middle
of the side đŽđ¶, which is đ”1.
This point has equal powers to the inscribed
circle (đŒ) and to the đ” âex-inscribed circle (đŒđ), so it
belongs to their radical axis.
Ion Patrascu, Florentin Smarandache
44
Analogously, đ¶1 is on the radical axis of the
circles (đŒ) and (đŒđ).
The radical axis of the circles (đŒ), (đŒđ ) is the
perpendicular taken from đ”1 to the bisector đŒđŒđ.
This bisector is parallel with the internal
bisector of the angle đŽ1đ”1đ¶1 , therefore the
perpendicular in đ”1 on đŒđŒđ is the external bisector of
the angle đŽ1đ”1đ¶1 from the median triangle.
Analogously, it follows that the radical axis of
the circles (đŒ), (đŒđ) is the external bisector of the angle
đŽ1đ¶1đ”1 from the median triangle.
Because the bisectors intersect in the center of
the circle đŽ1 -ex-inscribed to the median triangle
đŽ1đ”1đ¶1, this point đđ is the center of the radical center
of the circles (đŒ), (đŒđ), (đŒđ).
Remark.
The theorem for the circles (đŒ), (đŒđ), (đŒđ) and (đŒ),
(đŒđ ), (đŒđ ) can be proved analogously, obtaining the
points đđ and đđ.
3rd Theorem.
The radical circleâs radius of the circles ex-
inscribed to the triangle đŽđ”đ¶ is given by the formula: 1
2âđ2 + đ2, where đ is the radius of the inscribed circle.
Complements to Classic Topics of Circles Geometry
45
Proof.
The position vector of the circle đŒ of the
inscribed circle in the triangle ABC is:
đđŒ =1
2đ(đđđŽ + đđđ” + đđđ¶ ).
Spieckerâs point đ is the center of radical circle
of ex-inscribed circle and is the center of the inscribed
circle in the median triangle đŽ1đ”1đ¶1, therefore:
đđ =1
đ(1
2đđđŽ1 +
1
2đđđ”1 +
1
2đđđ¶1 ).
Figure 2.
We denote by đ the contact point with the đŽ-ex-
inscribed circle of the tangent taken from đ to this
circle (see Figure 2).
Ion Patrascu, Florentin Smarandache
46
The radical circleâs radius is given by:
đđ = âđđŒđ2 â đŒđ
2
đŒđđ =1
2đ(đđŒđđŽ1
+ đđŒđđ”1 + đđŒđđ¶1
).
We evaluate the product of the scales đŒđđ â đŒđđ ;
we have:
đŒđđ2 =
1
4đ2 (đ2đŒđđŽ12 + đ2đŒđđ”1
2 + đ2đŒđđ¶12 + 2đđđŒđđŽ1
â
đŒđđ”1 + 2đđđŒđđ”1
â đŒđđ¶1 + 2đđđŒđđŽ1
â đŒđđ¶1 ).
From the law of cosines applied in the triangle
đŒđđŽ1đ”1, we find that:
2đŒđđŽ1 â đŒđđ”1
= đŒđđŽ12 + đŒđđ”1
2 â1
4đ2, therefore:
2đđđŒđđŽ1 â đŒđđ”1
= đđ(đŒđđŽ12 + đŒđđ”1
2 â1
4đđđ2.
Analogously, we obtain:
2đđđŒđđ”1 â đŒđđ¶1
= đđ(đŒđđ”12 + đŒđđ¶1
2 â1
4đ2đđ,
2đđđŒđđŽ1 â đŒđđ¶1
= đđ(đŒđđŽ12 + đŒđđ¶1
2 â1
4đđ2đ.
đŒđđ2 =
1
4đ2[(đ2 + đđ + đđ)đŒđđŽ1
2 + (đ2 + đđ +
đđ)đŒđđ”12 + (đ2 + đđ + đđ)đŒđđ¶1
2 âđđđ
4(đ + đ + đ)],
đŒđđ2 =
1
4đ2 [2đ(đđŒđđŽ12 + đđŒđđ”1
2 + đđŒđđ¶12) â 2đ đđ],
đŒđđ2 =
1
2đ(đđŒđđŽ1
2 + đđŒđđ”12 + đđŒđđ¶1
2) â1
2đ đ.
From the right triangle đŒđđ·đđŽ1, we have that:
đŒđđŽ12 = đđ
2 + đŽ1đ·đ2 = đđ
2 + [đ
2â (đ â đ)]
2
=
= đđ2 +
(đâđ)2
4 .
From the right triangles đŒđđžđđ”1 Èi đŒđđčđđ¶1 , we
find:
Complements to Classic Topics of Circles Geometry
47
đŒđđ”12 = đđ
2 + đ”1đžđ2 = đđ
2 + [đ
2â (đ â đ)]
2
=
= đđ2 +
1
4(đ + đ)2,
đŒđđ¶12 = đđ
2 +1
4(đ + đ)2.
Evaluating đđŒđđŽ12 + đđŒđđ”1
2 + đđŒđđ¶12, we obtain:
đđŒđđŽ12 + đđŒđđ”1
2 + đđŒđđ¶12 =
= 2đđđ2 +
1
2đ(đđ + đđ + đđ) â
1
4đđđ.
But:
đđ + đđ + đđ = đ2 + đ2 + 4đ đ.
It follows that: 1
2đ[đđŒđđŽ1
2 + đđŒđđ”12 + đđŒđđ¶1
2] = đđ2 +
1
4(đ2 + đ2) +
1
2đ đ
and
đŒđđ2 = đđ
2 +1
4(đ2 + đ2).
Then, we obtain:
đđ =1
2âđ2 + đ2.
Ion Patrascu, Florentin Smarandache
48
References.
[1] C. Barbu: Teoreme fundamentale din geometria triunghiului [Fundamental Theorems of Triangle Geometry]. Bacau, Romania: Editura
Unique, 2008. [2] I. Patrascu, F. Smarandache: Variance on topics of
plane geometry. Columbus: The Educational
Publisher, Ohio, USA, 2013.
Complements to Classic Topics of Circles Geometry
49
The Polars of a Radical Center
In [1] , the late mathematician Cezar
Cosnita, using the barycenter coordinates,
proves two theorems which are the subject
of this article.
In a remark made after proving the first
theorem, C. Cosnita suggests an
elementary proof by employing the concept
of polar.
In the following, we prove the theorems
based on the indicated path, and state that
the second theorem is a particular case of
the former. Also, we highlight other
particular cases of these theorems.
1st Theorem.
Let đŽđ”đ¶ be a given triangle; through the pairs of
points (đ”, đ¶) , (đ¶, đŽ) and (đŽ, đ”) we take three circles
such that their radical center is on the outside.
The polar lines of the radical center of these
circles in relation to each of them cut the sides đ”đ¶, đ¶đŽ
and đŽđ” respectively in three collinear points.
Ion Patrascu, Florentin Smarandache
50
Proof.
We denote by đ·, đ, đč the second point of
intersection of the pairs of circles passing through
(đŽ, đ”) and (đ”, đ¶) ; (đ”, đ¶) and (đŽ, đ¶) , (đ”, đ¶) and (đŽ, đ”)
respectively (see Figure 1).
Figure 1
Let đ be the radical center of those circles. In
fact, {đ } = đŽđč â© đ”đ· â© đ¶đž.
We take from đ the tangents đ đ·1 = đ đ·2 to the
circle (đ”, đ¶), đ đž1 = đ đž2 to the circle (đŽ, đ¶) and đ đč1 =
đ đč2 to the circle passing through (đŽ, đ”). Actually, we
build the radical circle đ(đ , đ đ·1) of the given circles.
The polar lines of đ to these circles are the lines
đ·1đ·2, đž1đž2, đč1đč2. These three lines cut đ”đ¶, đŽđ¶ and đŽđ”
in the points đ, đ and đ , and these lines are
respectively the polar lines of đ in respect to the
Complements to Classic Topics of Circles Geometry
51
circles passing through (đ”, đ¶), (đ¶, đŽ) and (đŽ, đ”) . The
polar lines are the radical axis of the radical circle
with each of the circles passing through
(đ”, đ¶), (đ¶, đŽ), (đŽ, đ”), respectively. The points belong to
the radical axis having equal powers to those circles,
thereby đđ·1 â đđ·2 = đđ¶ â đđ”.
This relationship shows that the point đ has
equal powers relative to the radical circle and to the
circle circumscribed to the triangle đŽđ”đ¶; analogically,
the point đ has equal powers relative to the radical
circle and to the circle circumscribed to the triangle
đŽđ”đ¶ ; and, likewise, the point đ has equal powers
relative to the radical circle and to the circle
circumscribed to the triangle đŽđ”đ¶.
Because the locus of the points having equal
powers to two circles is generally a line, i.e. their
radical axis, we get that the points đ, đ and đ are
collinear, belonging to the radical axis of the radical
circle and to the circle circumscribed to the given
triangle.
2nd Theorem.
If đ is a point in the plane of the triangle đŽđ”đ¶
and the tangents in this point to the circles
circumscribed to triangles đ¶,đđŽđ¶, đđŽđ”, respectively,
cut đ”đ¶, đ¶đŽ and đŽđ”, respectively, in the points đ, đ, đ,
then these points are collinear.
Ion Patrascu, Florentin Smarandache
52
Proof.
The point đ is the radical center for the circles
(đđ”đ¶), (đđŽđ¶), and (đđŽđ”), and the tangents in đ to
these circles are the polar lines to đ in relation to
these circles.
If đ, đ, đ are the intersections of these tangents
(polar lines) with đ”đ¶, đ¶đŽ, đŽđ”, then they belong to the
radical axis of the circumscribed circle to the triangle
đŽđ”đ¶ and to the circle âreducedâ to the point đ (đđ2 =
đđ” â đđ¶, etc.).
Being located on the radical axis of the two
circles, the points đ, đ, đ are collinear.
Remarks.
1. Another elementary proof of this theorem is
to be found in [3].
2. If the circles from the 1st theorem are adjoint
circles of the triangle đŽđ”đ¶ , then they
intersect in Ω (the Brocardâs point).
Therefore, we get that the tangents taken in
Ω to the adjoin circles cut the sides đ”đ¶ , đ¶đŽ
and đŽđ” in collinear points.
Complements to Classic Topics of Circles Geometry
53
References.
[1] C. CoÈniÈÄ: CoordonĂ©es baricentrique [Barycentric Coordinates], Bucarest â Paris: Librairie Vuibert, 1941.
[3] I. PÄtraÈcu: Axe Èi centre radicale ale cercului adjuncte unui triunghi [Axis and radical centers of the adjoin circle of a triangle], in âRecreaÈii
matematiceâ, year XII, no. 1, 2010. [3] C. Mihalescu: Geometria elementelor remarcabile
[The Geometry of Outstanding Elements], Bucharest: Editura Tehnica, 1957.
[4] F. Smarandache, I. Patrascu: Geometry of
Homological Triangle, Columbus: The Educational Publisher Inc., 2012.
Ion Patrascu, Florentin Smarandache
54
Complements to Classic Topics of Circles Geometry
55
Regarding the First Droz-
Farnyâs Circle
In this article, we define the first Droz-
Farnyâs circle, we establish a connection
between it and a concyclicity theorem,
then we generalize this theorem, leading to
the generalization of Droz-Farnyâs circle.
The first theorem of this article was
enunciated by J. Steiner and it was proven
by Droz-Farny (Mathésis, 1901).
1st Theorem.
Let đŽđ”đ¶ be a triangle, đ» its orthocenter and
đŽ1, đ”1, đ¶1 the means of sides (đ”đ¶), (đ¶đŽ), (đŽđ”).
If a circle, having its center đ», intersects đ”1đ¶1 in
đ1, đ1 ; đ¶1đŽ1 in đ2, đ2 and đŽ1đ”1 in đ3, đ3 , then đŽđ1 =
đŽđ1 = đ”đ2 = đ”đ2 = đ¶đ3 = đ¶đ3.
Proof.
Naturally, đ»đ1 = đ»đ1, đ”1đ¶1 â„ đ”đ¶, đŽđ» â„ đ”đ¶.
It follows that đŽđ» â„ đ”1đ¶1.
Ion Patrascu, Florentin Smarandache
56
Figure 1.
Therefore, đŽđ» is the mediator of segment đ1đ1;
similarly, đ”đ» and đ¶đ» are the mediators of segments
đ2đ2 and đ3đ3.
Let đ1 be the intersection of lines đŽđ» and đ”1đ¶1
(see Figure 1); we have đ1đŽ2 â đ1đ»
2 = đ1đŽ2 â đ1đ»
2. We
denote đ đ» = đ»đ1. It follows that đ1đŽ2 = đ đ»
2 + (đ1đŽ +
đ1đ»)(đ1đŽ â đ1đ») = đ đ»2 + đŽđ» â (đ1đŽ â đ1đ»).
However, đ1đŽ = đ1đ»1, where đ»1 is the projection
of đŽ on đ”đ¶; we find that đ1đŽ2 = đ đ»
2 + đŽđ» â đ»đ»1.
It is known that the symmetric of orthocenter đ»
towards đ”đ¶ belongs to the circle of circumscribed
triangle đŽđ”đ¶.
Denoting this point by đ»1âČ , we have đŽđ» â đ»đ»1
âČ =
đ 2 â đđ»2 (the power of point đ» towards the
circumscribed circle).
Complements to Classic Topics of Circles Geometry
57
We obtain that đŽđ» â đ»đ»1 =1
2â (đ 2 â đđ»2) , and
therefore đŽđ12 = đ đ»
2 +1
2â (đ 2 â đđ»2) , where đ is the
center of the circumscribed triangle đŽđ”đ¶.
Similarly, we find đ”đ22 = đ¶đ3
2 = đ đ»2 +
1
2(đ 2 â
đđ»2), therefore đŽđ1 = đ”đ2 = đ¶đ3.
Remarks.
a. The proof adjusts analogously for the
obtuse triangle.
b. 1st Theorem can be equivalently
formulated in this way:
2nd Theorem.
If we draw three congruent circles centered in a
given triangleâs vertices, the intersection points of
these circles with the sides of the median triangle of
given triangle (middle lines) are six points situated on
a circle having its center in triangleâs orthocenter.
If we denote by đ the radius of three congruent
circles having đŽ, đ”, đ¶ as their centers, we get:
đ đ»2 = đ2 +
1
2(đđ»2 â đ 2).
However, in a triangle, đđ»2 = 9đ 2 â (đ2 + đ2 +
đ2), đ being the radius of the circumscribed circle; it
follows that:
đ đ»2 = đ2 + 4đ 2 â
1
2(đ2 + đ2 + đ2).
Ion Patrascu, Florentin Smarandache
58
Remark.
A special case is the one in which đ = đ , where
we find that đ đ»2 = đ 1
2 = 5đ 2 â1
2(đ2 + đ2 + đ2) =
1
2(đ 2 +
đđ»2).
Definition.
The circle đ(đ», đ 1), where:
đ 1 = â5đ 2 â1
2(đ2 + đ2 + đ2),
is called the first Droz-Farnyâs circle of the triangle
đŽđ”đ¶.
Remark.
Another way to build the first Droz-Farnyâs circle
is offered by the following theorem, which, according
to [1], was the subject of a problem proposed in 1910
by V. Thébault in the Journal de Mathématiques
Elementaire.
3rd Theorem.
The circles centered on the feet of a triangleâs
altitudes passing through the center of the circle
circumscribed to the triangle cut the triangleâs sides
in six concyclical points.
Complements to Classic Topics of Circles Geometry
59
Proof.
We consider đŽđ”đ¶ an acute triangle and đ»1, đ»2, đ»3
the altitudesâ feet. We denote by đŽ1, đŽ2; đ”1, đ”2; đ¶1, đ¶2
the intersection points of circles having their centers
đ»1, đ»2, đ»3 to đ”đ¶, đ¶đŽ, đŽđ”, respectively.
We calculate đ»đŽ2 of the right angled triangle
đ»đ»1đŽ2 (see Figure 2). We have đ»đŽ22 = đ»đ»1
2 + đ»1đŽ22.
Because đ»1đŽ2 = đ»1đ , it follows that đ»đŽ22 =
đ»đ»12 + đ»1đ
2 . We denote by đ9 the mean of segment
đđ» ; the median theorem in triangle đ»1đ»đ leads to
đ»1đ»2 + đ»1đ
2 = 2đ»1đ92 +
1
2đđ»2.
It is known that đ9đ»1 is the nine-points circleâs
radius, so đ»1đ9 =1
2đ ; we get: đ»đŽ1
2 =1
2(đ 2 + đđ»2) ;
similarly, we find that đ»đ”12 = đ»đ¶1
2 =1
2(đ 2 + đđ»2) ,
which shows that the points đŽ1, đŽ2; đ”1, đ”2; đ¶1, đ¶2
belong to the first Droz-Farnyâs circle.
Figure 2.
Ion Patrascu, Florentin Smarandache
60
Remark.
The 2nd and the 3rd theorems show that the first
Droz-Farnyâs circle pass through 12 points, situated
two on each side of the triangle and two on each side
of the median triangle of the given triangle.
The following theorem generates the 3rd Theorem.
4th Theorem.
The circles centered on the feet of altitudes of a
given triangle intersect the sides in six concyclical
points if and only if their radical center is the center
of the circle circumscribed to the given triangle.
Proof.
Let đŽ1, đŽ2; đ”1, đ”2; đ¶1, đ¶2 be the points of
intersection with the sides of triangle đŽđ”đ¶ of circles
having their centers in altitudesâ feet đ»1, đ»2, đ»3.
Suppose the points are concyclical; it follows
that their circleâs radical center and of circles centered
in đ»2 and đ»3 is the point đŽ (sides đŽđ” and đŽđ¶ are
radical axes), therefore the perpendicular from đŽ on
đ»2đ»3 is radical axis of centers having their centers đ»2
and đ»3.
Since đ»2đ»3 is antiparallel to đ”đ¶, it is parallel to
tangent taken in đŽ to the circle circumscribed to
triangle đŽđ”đ¶.
Complements to Classic Topics of Circles Geometry
61
Consequently, the radical axis is the
perpendicular taken in đŽ on the tangent to the
circumscribed circle, therefore it is đŽđ.
Similarly, the other radical axis of circles
centered in đ»1, đ»2 and of circles centered in đ»1, đ»3 pass
through đ, therefore đ is the radical center of these
circles.
Reciprocally.
Let đ be the radical center of the circles having
their centers in the feet of altitudes. Since đŽđ is
perpendicular on đ»2đ»3 , it follows that đŽđ is the
radical axis of circles having their centers in đ»2, đ»3,
therefore đŽđ”1 â đŽđ”2 = đŽđ¶1 â đŽđ¶2.
From this relationship, it follows that the points
đ”1, đ”2; đ¶1, đ¶2 are concyclic; the circle on which these
points are located has its center in the orthocenter đ»
of triangle đŽđ”đ¶.
Indeed, the mediatorsâ chords đ”1đ”2 and đ¶1đ¶2 in
the two circles are the altitudes from đ¶ and đ” of
triangle đŽđ”đ¶, therefore đ»đ”1 = đ»đ”2 = đ»đ¶1 = đ»đ¶2.
This reasoning leads to the conclusion that đ”đ is
the radical axis of circles having their centers đ»1 and
đ»3 , and from here the concyclicality of the points
đŽ1, đŽ2; đ¶1, đ¶2 on a circle having its center in đ» ,
therefore đ»đŽ1 = đ»đŽ2 = đ»đ¶1 = đ»đ¶2 . We obtained that
đ»đŽ1 = đ»đŽ2 = đ»đ”1 = đ»đ”2 = đ»đ¶1 = đ»đ¶2 , which shows
the concyclicality of points đŽ1, đŽ2, đ”1, đ”2, đ¶1, đ¶2.
Ion Patrascu, Florentin Smarandache
62
Remark.
The circles from the 3rd Theorem, passing
through đ and having noncollinear centers, admit đ
as radical center, and therefore the 3rd Theorem is a
particular case of the 4th Theorem.
References.
[1] N. Blaha: Asupra unor cercuri care au ca centre douÄ puncte inverse [On some circles that have
as centers two inverse points], in âGazeta Matematicaâ, vol. XXXIII, 1927.
[2] R. Johnson: Advanced Euclidean Geometry. New York: Dover Publication, Inc. Mineola, 2004
[3] Eric W. Weisstein: First Droz-Farnyâs circle. From
Math World â A Wolfram WEB Resurse, http://mathworld.wolfram.com/.
[4] F. Smarandache, I. Patrascu: Geometry of
Homological Triangle. Columbus: The Education Publisher Inc., Ohio, SUA, 2012.
Complements to Classic Topics of Circles Geometry
63
Regarding the Second Droz-
Farnyâs Circle
In this article, we prove the theorem
relative to the second Droz-Farnyâs circle,
and a sentence that generalizes it.
The paper [1] informs that the following
Theorem is attributed to J. Neuberg
(Mathesis, 1911).
1st Theorem.
The circles with its centers in the middles of
triangle đŽđ”đ¶ passing through its orthocenter đ»
intersect the sides đ”đ¶, đ¶đŽ and đŽđ” respectively in the
points đŽ1, đŽ2, đ”1, đ”2 and đ¶1, đ¶2, situated on a concentric
circle with the circle circumscribed to the triangle đŽđ”đ¶
(the second Droz-Farnyâs circle).
Proof.
We denote by đ1, đ2, đ3 the middles of đŽđ”đ¶
triangleâs sides, see Figure 1. Because đŽđ» â„ đ2đ3 and
đ» belongs to the circles with centers in đ2 and đ3, it
follows that đŽđ» is the radical axis of these circles,
Ion Patrascu, Florentin Smarandache
64
therefore we have đŽđ¶1 â đŽđ¶2 = đŽđ”2 â đŽđ”1. This relation
shows that đ”1, đ”2, đ¶1, đ¶2 are concyclic points, because
the center of the circle on which they are situated is đ,
the center of the circle circumscribed to the triangle
đŽđ”đ¶, hence we have that:
đđ”1 = đđ¶1 = đđ¶2 = đđ”2. (1)
Figure 1.
Analogously, đ is the center of the circle on
which the points đŽ1, đŽ2, đ¶1, đ¶2 are situated, hence:
đđŽ1 = đđ¶1 = đđ¶2 = đđŽ2. (2)
Also, đ is the center of the circle on which the
points đŽ1, đŽ2, đ”1, đ”2 are situated, and therefore:
đđŽ1 = đđ”1 = đđ”2 = đđŽ2. (3)
The relations (1), (2), (3) show that the points
đŽ1, đŽ2, đ”1, đ”2, đ¶1, đ¶2 are situated on a circle having the
center in đ, called the second Droz-Farnyâs circle.
Complements to Classic Topics of Circles Geometry
65
1st Proposition.
The radius of the second Droz-Farnyâs circle is
given by:
đ 22 = 5đ2 â
1
2(đ2 + đ2 + đ2).
Proof.
From the right triangle đđ1đŽ1, using Pitagoraâs
theorem, it follows that:
đđŽ12 = đđ1
2 + đŽ1đ12 = đđ1
2 + đ1đ2.
From the triangle đ”đ»đ¶ , using the median
theorem, we have:
đ»đ12 =
1
4[2(đ”đ»2 + đ¶đ»2) â đ”đ¶2].
But in a triangle,
đŽđ» = 2đđ1, đ”đ» = 2đđ2, đ¶đ» = 2đđ3,
hence:
đ»đ12 = 2đđ2
2 + 2đđ32 =
đ2
4.
But:
đđ12 = đ 2 â
đ2
4 ;
đđ22 = đ 2 â
đ2
4 ;
đđ32 = đ 2 â
đ2
4 ,
where R is the radius of the circle circumscribed to the
triangle đŽđ”đ¶.
We find that đđŽ12 = đ 2
2 = 5đ 2 â1
2(đ2 + đ2 + đ2).
Ion Patrascu, Florentin Smarandache
66
Remarks.
a. We can compute đđ12 + đ1đ2 using the
median theorem in the triangle đđ1đ» for
the median đ1đ9 (đ9 is the center of the
nine points circle, i.e. the middle of (đđ»)).
Because đ9đ1 =1
2đ , we obtain: đ 2
2 =
1
2(đđ2 + đ 2). In this way, we can prove
the Theorem computing đđ”12 and đđ¶1
2.
b. The statement of the 1st Theorem was the
subject no. 1 of the 49th International
Olympiad in Mathematics, held at Madrid
in 2008.
c. The 1st Theorem can be proved in the same
way for an obtuse triangle; it is obvious
that for a right triangle, the second Droz-
Farnyâs circle coincides with the circle
circumscribed to the triangle đŽđ”đ¶.
d. The 1st Theorem appears as proposed
problem in [2].
2nd Theorem.
The three pairs of points determined by the
intersections of each circle with the center in the
middle of triangleâs side with the respective side are
on a circle if and only these circles have as radical
center the triangleâs orthocenter.
Complements to Classic Topics of Circles Geometry
67
Proof.
Let đ1, đ2, đ3 the middles of the sides of triangle
đŽđ”đ¶ and let đŽ1, đŽ2, đ”1, đ”2, đ¶1, đ¶2 the intersections with
đ”đ¶, đ¶đŽ, đŽđ” respectively of the circles with centers in
đ1, đ2, đ3.
Let us suppose that đŽ1, đŽ2, đ”1, đ”2, đ¶1, đ¶2 are
concyclic points. The circle on which they are situated
has evidently the center in đ, the center of the circle
circumscribed to the triangle đŽđ”đ¶.
The radical axis of the circles with centers đ2, đ3
will be perpendicular on the line of centers đ2đ3, and
because A has equal powers in relation to these circles,
since đŽđ”1 â đŽđ”2 = đŽđ¶1 â đŽđ¶2, it follows that the radical
axis will be the perpendicular taken from A on đ2đ3,
i.e. the height from đŽ of triangle đŽđ”đ¶.
Furthermore, it ensues that the radical axis of
the circles with centers in đ1 and đ2 is the height
from đ” of triangle đŽđ”đ¶ and consequently the
intersection of the heights, hence the orthocenter đ» of
the triangle đŽđ”đ¶ is the radical center of the three
circles.
Reciprocally.
If the circles having the centers in đ1, đ2, đ3
have the orthocenter with the radical center, it follows
that the point đŽ, being situated on the height from A
which is the radical axis of the circles of centers đ2, đ3
Ion Patrascu, Florentin Smarandache
68
will have equal powers in relation to these circles and,
consequently, đŽđ”1 â đŽđ”2 = đŽđ¶1 â đŽđ¶2 , a relation that
implies that đ”1, đ”2, đ¶1, đ¶2 are concyclic points, and the
circle on which these points are situated has đ as its
center.
Similarly, đ”đŽ1 â đ”đŽ2 = đ”đ¶1 â đ”đ¶2 , therefore
đŽ1, đŽ2, đ¶1, đ¶2 are concyclic points on a circle of center đ.
Having đđ”1 = đđ”2 = đđ¶1 = đđ¶2 and đđŽ1 â đđŽ2 = đđ¶1 â
đđ¶2 , we get that the points đŽ1, đŽ2, đ”1, đ”2, đ¶1, đ¶2 are
situated on a circle of center đ.
Remarks.
1. The 1st Theorem is a particular case of the
2nd Theorem, because the three circles of
centers đ1, đ2, đ3 pass through đ», which
means that đ» is their radical center.
2. The Problem 525 from [3] leads us to the
following Proposition providing a way to
construct the circles of centers đ1, đ2, đ3
intersecting the sides in points that
belong to a Droz-Farnyâs circle of type 2.
2nd Proposition.
The circles â (đ1,1
2âđ + đ2), â (đ2,
1
2âđ + đ2),
â (đ3,1
2âđ + đ2) intersect the sides đ”đ¶ , đ¶đŽ , đŽđ”
respectively in six concyclic points; đ is a conveniently
Complements to Classic Topics of Circles Geometry
69
chosen constant, and đ, đ, đ are the lengths of the sides
of triangle đŽđ”đ¶.
Proof.
According to the 2nd Theorem, it is necessary to
prove that the orthocenter đ» of triangle đŽđ”đ¶ is the
radical center for the circles from hypothesis.
Figure 2.
The power of đ» in relation with
â (đ1,1
2âđ + đ2) is equal to đ»đ1
2 â1
4(đ + đ2) . We
observed that đ12 = 4đ 2 â
đ2
2â
đ2
2â
đ2
4 , therefore
đ»đ12 â
1
4(đ + đ2) = 4đ 2 â
đ2+đ2+đ2
4â
1
4đ. We use the
same expression for the power of H in relation to the
circles of centers đ2, đ3, hence H is the radical center
of these three circles.
Ion Patrascu, Florentin Smarandache
70
References.
[1] C. Mihalescu: Geometria elementelor remarcabile [The Geometry of Outstanding Elements]. Bucharest: Editura TehnicÄ, 1957.
[2] I. PÄtraĆcu: Probleme de geometrie planÄ [Some Problems of Plane Geometry], Craiova: Editura Cardinal, 1996.
[3] C. CoĆniĆŁÄ: Teoreme Ći probleme alese de matematicÄ [Theorems and Problems],
BucureĆti: Editura de Stat DidacticÄ Ći PedagogicÄ, 1958.
[4] I. PÄtraÈcu, F. Smarandache: Variance on Topics of
Plane Geometry, Educational Publishing, Columbus, Ohio, SUA, 2013.
Complements to Classic Topics of Circles Geometry
71
Neubergâs Orthogonal Circles
In this article, we highlight some metric
properties in connection with Neuberg's
circles and triangle.
We recall some results that are necessary.
1st Definition.
It's called Brocardâs point of the triangle đŽđ”đ¶ the
point Ω with the property: âąÎ©đŽđ” = âąÎ©đ”đ¶ = âąÎ©đ¶đŽ .
The measure of the angle ΩđŽđ” is denoted by đ and it
is called Brocard's angle. It occurs the relationship:
ctgđ = ctgđŽ + ctgđ” + ctgđ¶ (see [1]).
Figure 1.
Ion Patrascu, Florentin Smarandache
72
2nd Definition.
Two triangles are called equibrocardian if they
have the same Brocardâs angle.
3rd Definition.
The locus of points đ from the plane of the
triangle located on the same side of the line đ”đ¶ as đŽ
and forming with đ”đ¶ an equibrocardian triangle with
đŽđ”đ¶, containing the vertex đŽ of the triangle, it's called
A-Neubergâ circle of the triangle đŽđ”đ¶.
We denote by đđ the center of A-Neubergâ circle
by radius đđ (analogously, we define B-Neubergâ and
C-Neubergâ circles).
We get that đ(đ”đđđ¶) = 2đ and đđ =đ
2âctg2đ â 3
(see [1]).
The triangle đđđđđđ formed by the centers of
Neubergâs circles is called Neubergâs triangle.
1st Proposition.
The distances from the center circumscribed to
the triangle đŽđ”đ¶ to the vertex of Neubergâs triangle
are proportional to the cubes of đŽđ”đ¶ triangleâs sides
lengths.
Complements to Classic Topics of Circles Geometry
73
Proof.
Let đ be the center of the circle circumscribed to
the triangle đŽđ”đ¶ (see Figure 2).
Figure 2.
The law of cosines applied in the triangle đđđđ”
provides: đđđ
sin(đđđ”đ)=
đ
sinđ .
But đ(âąđđđ”đ) = đ(âąđđđ”đ¶) â đ(âąđđ”đ¶) = đŽ â đ.
We have that đđđ
sin(đŽâđ)=
đ
sinđ .
But
sin(đŽâđ)
sinđ=
đ¶Î©
AΩ=
đ
đâ2đ sinđ
đ
đâ2đ sinđ
=đ3
đđđ=
đ3
4đ đ ,
đ being the area of the triangle đŽđ”đ¶.
Ion Patrascu, Florentin Smarandache
74
It follows that đđđ =đ3
4đ , and we get that
đđđ
đ3 =
đđđ
đ3 =đđđ
đ3 .
Consequence.
In a triangle ABC, we have:
1) đđđ â đđđ â đđđ = đ 3;
2) ctgđ =đđđ
đ+
đđđ
đ+
đđđ
đ .
2nd Proposition.
If đđđđđđ is the Neubergâs triangle of the
triangle đŽđ”đ¶, then:
đđđđ2 =
(đ2 + đ2)(đ4 + đ4) â đ2đ2đ2
2đ2đ2 + 2đ2đ2 + 2đ2đ2 â đ4 â đ4 â đ4 .
(The formulas for đđđđ and đđđđ are obtained
from the previous one, by circular permutations.)
Proof.
We apply the law of cosines in the triangle
đđđđđ:
đđđ =đ3
4đ , đđđ =
đ3
4đ ,đ(âąđđđđđ) = 1800 â ïżœïżœ.
đđđđ2 =
đ6 + đ6 â 2đ3đ3cos (1800 â đ)
16đ2
=đ6 + đ6 + 2đ3đ3cosđ¶
16đ2 .
Complements to Classic Topics of Circles Geometry
75
But the law of cosines in the triangle đŽđ”đ¶
provides
2cosđ¶ =đ2 + đ2 â đ2
2đđ
and, from din Heronâs formula, we find that
16đ2 = 2đ2đ2 + 2đ2đ2 + 2đ2đ2 â đ4 â đ4 â đ4.
Substituting the results above, we obtain, after a
few calculations, the stated formula.
4th Definition.
Two circles are called orthogonal if they are
secant and their tangents in the common points are
perpendicular.
3rd Proposition. (Gaultier â 18B)
Two circles đ(đ1, đ1), đ(đ2, đ2) are orthogonal if
and only if
đ12 + đ2
2 = đ1đ22.
Proof.
Let đ(đ1, đ1), đ(đ2, đ2) be orthogonal (see Figure
3); then, if đŽ is one of the common points, the triangle
đ1đŽđ2 is a right triangle and the Pythagorean Theorem
applied to it, leads to đ12 + đ2
2 = đ1đ22.
Ion Patrascu, Florentin Smarandache
76
Reciprocally.
If the metric relationship from the statement
occurs, it means that the triangle đ1đŽđ2 is a right
triangle, therefore đŽ is their common point (the
relationship đ12 + đ2
2 = đ1đ22 implies đ1
2 + đ22 >
đ1đ22), then đ1đŽ â„ đ2đŽ, so đ1đŽ is tangent to the circle
đ(đ2, đ2) because it is perpendicular in đŽ on radius đ2đŽ,
and as well đ2đŽ is tangent to the circle đ(đ1, đ1) ,
therefore the circles are orthogonal.
Figure 3.
4th Proposition.
B-Neubergâs and C-Neubergâs circles associated
to the right triangle đŽđ”đ¶ (in đŽ) are orthogonal.
Complements to Classic Topics of Circles Geometry
77
Proof.
If đ(ïżœïżœ) = 900, then đđđđ2 =
đ6+đ6
16đ.
đđ =đ
2âctg2đ â 3 ; đđ =
đ
2âctg2đ â 3.
But ctgđ âđ2+đ2+đ2
4đ=
đ2+đ2
2đ=
đ2
đđ .
It was taken into account that đ2 = đ2 + đ2 and
2đ = đđ.
ctg2đ â 3 =đ4
đ2đ2â 3 =
(đ2 + đ2)2 â 3đ2đ2
đ2đ2
ctg2đ â 3 =đ4 + đ4 â đ2đ2
đ2đ2
đđ2 + đđ
2 =đ4 + đ4 â đ2đ2
đ2đ2(đ2 + đ2
4)
=(đ2 + đ2)(đ4 + đ4 â đ2đ2)
4đ2đ2=
đ6 + đ6
16đ2 .
By đđ2 + đđ
2 = đđđđ2, it follows that B-Neubergâs
and C-Neubergâs circles are orthogonal.
Ion Patrascu, Florentin Smarandache
78
References.
[1] F. Smarandache, I. Patrascu: The Geometry of Homological Triangles. Columbus: The Educational Publisher, Ohio, USA, 2012.
[2] T. Lalescu: Geometria triunghiului [The Geometry of the Triangle]. Craiova: Editura Apollo, 1993.
[3] R. A. Johnson: Advanced Euclidian Geometry. New
York: Dover Publications, 2007.
Complements to Classic Topics of Circles Geometry
79
Lucasâs Inner Circles
In this article, we define the Lucasâs inner
circles and we highlight some of their
properties.
1. Definition of the Lucasâs Inner Circles
Let đŽđ”đ¶ be a random triangle; we aim to
construct the square inscribed in the triangle đŽđ”đ¶ ,
having one side on đ”đ¶.
Figure 1.
In order to do this, we construct a square đŽ,đ”,đ¶ ,đ·,
with đŽ, â (đŽđ”), đ”,, đ¶ , â (đ”đ¶) (see Figure 1).
We trace the line đ”đ·, and we note with đ·đ its
intersection with (đŽđ¶) ; through đ·đ we trace the
Ion Patrascu, Florentin Smarandache
80
parallel đ·đđŽđ to đ”đ¶ with đŽđ â (đŽđ”) and we project
onto đ”đ¶ the points đŽđ, đ·đ in đ”đ respectively đ¶đ.
We affirm that the quadrilateral đŽđđ”đđ¶đđ·đ is the
required square.
Indeed, đŽđđ”đđ¶đđ·đ is a square, because đ·đđ¶đ
đ·,đ¶ , =
đ”đ·đ
đ”đ·, =đŽđđ·đ
đŽ,đ·, and, as đ·,đ¶ , = đŽ,đ·, , it follows that đŽđđ·đ =
đ·đđ¶đ.
Definition.
It is called A-Lucasâs inner circle of the triangle
đŽđ”đ¶ the circle circumscribed to the triangle AAaDa.
We will note with đżđ the center of the A-Lucasâs
inner circle and with đđ its radius.
Analogously, we define the B-Lucasâs inner circle
and the C-Lucasâs inner circle of the triangle đŽđ”đ¶.
2. Calculation of the Radius of
the A-Lucas Inner Circle
We note đŽđđ·đ = đ„, đ”đ¶ = đ; let âđ be the height
from đŽ of the triangle đŽđ”đ¶.
The similarity of the triangles đŽđŽđđ·đ and đŽđ”đ¶
leads to: đ„
đ=
âđâđ„
âđ, therefore đ„ =
đâđ
đ+âđ .
From đđ
đ =
đ„
đ we obtain đđ =
đ .âđ
đ+âđ . (1)
Complements to Classic Topics of Circles Geometry
81
Note.
Relation (1) and the analogues have been
deduced by Eduard Lucas (1842-1891) in 1879 and
they constitute the âbirth certificate of the Lucasâs
circlesâ.
1st Remark.
If in (1) we replace âđ =2đ
đ and we also keep into
consideration the formula đđđ = 4đ đ, where đ is the
radius of the circumscribed circle of the triangle đŽđ”đ¶
and đ represents its area, we obtain:
đđ =đ
1+2đđ
đđ
[see Ref. 2].
3. Properties of the Lucasâs Inner Circles
1st Theorem.
The Lucasâs inner circles of a triangle are inner
tangents of the circle circumscribed to the triangle and
they are exteriorly tangent pairwise.
Proof.
The triangles đŽđŽđđ·đ and đŽđ”đ¶ are homothetic
through the homothetic center đŽ and the rapport: âđ
đ+âđ.
Ion Patrascu, Florentin Smarandache
82
Because đđ
đ =
âđ
đ+âđ, it means that the A-Lucasâs inner
circle and the circle circumscribed to the triangle đŽđ”đ¶
are inner tangents in đŽ.
Analogously, it follows that the B-Lucasâs and C-
Lucasâs inner circles are inner tangents of the circle
circumscribed to đŽđ”đ¶.
Figure 2.
We will prove that the A-Lucasâs and C-Lucasâs
circles are exterior tangents by verifying
đżđđżđ = đđ + đđ. (2)
We have:
đđżđ = đ â đđ;
đđżđ = đ â đđ
and
đ(đŽ0ïżœïżœ) = 2đ”
(if đ(ïżœïżœ) > 90° then đ(đŽ0ïżœïżœ) = 360° â 2đ”).
Complements to Classic Topics of Circles Geometry
83
The theorem of the cosine applied to the triangle
đđżđđżđ implies, keeping into consideration (2), that:
(đ â đđ)2 + (đ â đđ)
2 â 2(đ â đđ)(đ â đđ)đđđ 2đ” =
= (đđ + đđ)2.
Because đđđ 2đ” = 1 â 2đ đđ2đ”, it is found that (2)
is equivalent to:
đ đđ2đ” =đđđđ
(đ âđđ)(đ âđđ) . (3)
But we have: đđđđ =đ 2đđ2đ
(2đđ +đđ)(2đđ +đđ) ,
đđ + đđ = đ đ(đ
2đđ +đđ+
đ
2đđ +đđ).
By replacing in (3), we find that đ đđ 2đ” =đđ2đ
4đđđ 2 =
đ2
4đ2 âș sinđ” =đ
2đ is true according to the sines theorem.
So, the exterior tangent of the A-Lucasâs and C-Lucasâs
circles is proven.
Analogously, we prove the other tangents.
2nd Definition.
It is called an A-Apolloniusâs circle of the random
triangle đŽđ”đ¶ the circle constructed on the segment
determined by the feet of the bisectors of angle đŽ as
diameter.
Remark.
Analogously, the B-Apolloniusâs and C-
Apolloniusâs circles are defined. If đŽđ”đ¶ is an isosceles
triangle with đŽđ” = đŽđ¶ then the A-Apolloniusâs circle
Ion Patrascu, Florentin Smarandache
84
isnât defined for đŽđ”đ¶ , and if đŽđ”đ¶ is an equilateral
triangle, its Apolloniusâs circle isnât defined.
2nd Theorem.
The A-Apolloniusâs circle of the random triangle
is the geometrical point of the points đ from the plane
of the triangle with the property: đđ”
đđ¶=
đ
đ.
3rd Definition.
We call a fascicle of circles the bunch of circles
that do not have the same radical axis.
a. If the radical axis of the circlesâ fascicle is
exterior to them, we say that the fascicle
is of the first type.
b. If the radical axis of the circlesâ fascicle is
secant to the circles, we say that the
fascicle is of the second type.
c. If the radical axis of the circlesâ fascicle is
tangent to the circles, we say that the
fascicle is of the third type.
3rd Theorem.
The A-Apolloniusâs circle and the B-Lucasâs and
C-Lucasâs inner circles of the random triangle đŽđ”đ¶
form a fascicle of the third type.
Complements to Classic Topics of Circles Geometry
85
Proof.
Let {đđŽ} = đżđđżđ â© đ”đ¶ (see Figure 3).
Menelausâs theorem applied to the triangle đđ”đ¶
implies that: đđŽđ”
đđŽđ¶.đżđđ”
đżđđ.đżđđ
đżđđ¶= 1,
so:
đđŽđ”
đđŽđ¶.
đđ
đ âđđ.đ âđđ
đđ= 1
and by replacing đđ and đđ, we find that: đđŽđ”
đđŽđ¶=
đ2
đ2.
This relation shows that the point đđŽ is the foot
of the exterior symmedian from đŽ of the triangle đŽđ”đ¶
(so the tangent in đŽ to the circumscribed circle),
namely the center of the A-Apolloniusâs circle.
Let đ1 be the contact point of the B-Lucasâs and
C-Lucasâs circles. The radical center of the B-Lucasâs,
C-Lucasâs circles and the circle circumscribed to the
triangle đŽđ”đ¶ is the intersection đđŽ of the tangents
traced in đ” and in đ¶ to the circle circumscribed to the
triangle đŽđ”đ¶.
It follows that đ”đđŽ = đ¶đđŽ = đ1đđŽ, so đ1 belongs to
the circle đđŽ that has the center in đđŽ and orthogonally
cuts the circle circumscribed in đ” and đ¶. The radical
axis of the B-Lucasâs and C-Lucasâs circles is đđŽđ1, and
đđŽđ1 is tangent in đ1 to the circle đđŽ. Considering the
power of the point đđŽ in relation to đđŽ, we have:
đđŽđ12 = đđŽđ”. đđŽđ¶.
Ion Patrascu, Florentin Smarandache
86
Figure 3.
Also, đđŽđ2 = đđŽđ” â đđŽđ¶ ; it thus follows that
đđŽđŽ = đđŽđ1 , which proves that đ1 belongs to the A-
Apolloniusâs circle and is the radical center of the A-
Apolloniusâs, B-Lucasâs and C-Lucasâs circles.
Remarks.
1. If the triangle đŽđ”đ¶ is right in đŽ then
đżđđżđ||đ”đ¶, the radius of the A-Apolloniusâs
circle is equal to: đđđ
|đ2âđ2|. The point đ1 is
the foot of the bisector from đŽ. We find
that đđŽđ1 =đđđ
|đ2âđ2|, so the theorem stands
true.
Complements to Classic Topics of Circles Geometry
87
2. The A-Apolloniusâs and A-Lucasâs circles
are orthogonal. Indeed, the radius of the
A-Apolloniusâs circle is perpendicular to
the radius of the circumscribed circle, đđŽ,
so, to the radius of the A-Lucasâs circle
also.
4th Definition.
The triangle đđŽđđ”đđ¶ determined by the tangents
traced in đŽ, đ”, đ¶ to the circle circumscribed to the
triangle đŽđ”đ¶ is called the tangential triangle of the
triangle đŽđ”đ¶.
1st Property.
The triangle đŽđ”đ¶ and the Lucasâs triangle đżđđżđđżđ
are homological.
Proof.
Obviously, đŽđżđ, đ”đżđ , đ¶đżđ are concurrent in đ ,
therefore đ, the center of the circle circumscribed to
the triangle đŽđ”đ¶, is the homology center.
We have seen that {đđŽ} = đżđđżđ â© đ”đ¶ and đđŽ is the
center of the A-Apolloniusâs circle, therefore the
homology axis is the Apolloniusâs line đđŽđđ”đđ¶ (the line
determined by the centers of the Apolloniusâs circle).
Ion Patrascu, Florentin Smarandache
88
2nd Property.
The tangential triangle and the Lucasâs triangle
of the triangle đŽđ”đ¶ are orthogonal triangles.
Proof.
The line đđŽđ1 is the radical axis of the B-Lucasâs
inner circle and the C-Lucasâs inner circle, therefore it
is perpendicular on the line of the centers đżđđżđ .
Analogously, đđ”đ2 is perpendicular on đżđđżđ , because
the radical axes of the Lucasâs circles are concurrent
in đż, which is the radical center of the Lucasâs circles;
it follows that đđŽđđ”đđ¶ and đżđđżđđżđ are orthological and
đż is the center of orthology. The other center of
orthology is đ the center of the circle circumscribed to
đŽđ”đ¶.
References.
[1] D. BrĂąnzei, M. MiculiÈa. Lucas circles and spheres. In âThe Didactics of Mathematicsâ, volume 9/1994, Cluj-Napoca (Romania), p. 73-80.
[2] P. Yiu, A. P. Hatzipolakis. The Lucas Circles of a Triangle. In âAmer. Math. Monthlyâ, no.
108/2001, pp. 444-446. http://www.math.fau. edu/yiu/monthly437-448.pdf.
[3] F. Smarandache, I. Patrascu. The Geometry of
Homological Triangles. Educational Publisher, Columbus, Ohio, U.S.A., 2013.
Complements to Classic Topics of Circles Geometry
89
Theorems with Parallels Taken
through a Triangleâs Vertices
and Constructions Performed
only with the Ruler
In this article, we solve problems of
geometric constructions only with the
ruler, using known theorems.
1st Problem.
Being given a triangle đŽđ”đ¶ , its circumscribed
circle (its center known) and a point đ fixed on the
circle, construct, using only the ruler, a transversal
line đŽ1, đ”1, đ¶1, with đŽ1 â đ”đ¶, đ”1 â đ¶đŽ, đ¶1 â đŽđ”, such that
âąđđŽ1đ¶ ⥠âąđđ”1đ¶ ⥠âąđđ¶1đŽ (the lines taken though đ
to generate congruent angles with the sides đ”đ¶ , đ¶đŽ
and đŽđ”, respectively).
2nd Problem.
Being given a triangle đŽđ”đ¶ , its circumscribed
circle (its center known) and đŽ1, đ”1, đ¶1, such that đŽ1 â
Ion Patrascu, Florentin Smarandache
90
đ”đ¶, đ”1 â đ¶đŽ, đ¶1 â đŽđ” and đŽ1, đ”1, đ¶1 collinear, construct,
using only the ruler, a point đ on the circle
circumscribing the triangle, such that the lines
đđŽ1, đđ”1, đđ¶1 to generate congruent angles with đ”đ¶,
đ¶đŽ and đŽđ”, respectively.
3rd Problem.
Being given a triangle đŽđ”đ¶ inscribed in a circle
of given center and đŽđŽâČ a given cevian, đŽâČ a point on
the circle, construct, using only the ruler, the isogonal
cevian đŽđŽ1 to the cevian đŽđŽâČ.
To solve these problems and to prove the
theorems for problems solving, we need the following
Lemma:
1st Lemma. (Generalized Simpson's Line)
If đ is a point on the circle circumscribed to the
triangle đŽđ”đ¶ and we take the lines đđŽ1, đđ”1, đđ¶1
which generate congruent angles ( đŽ1 â đ”đ¶, đ”1 â
đ¶đŽ, đ¶1 â đŽđ”) with đ”đ¶, đ¶đŽ and đŽđ” respectively, then the
points đŽ1, đ”1, đ¶1 are collinear.
Complements to Classic Topics of Circles Geometry
91
Proof.
Let đ on the circle circumscribed to the triangle
đŽđ”đ¶ (see Figure 1), such that:
âąđđŽ1đ¶ ⥠âąđđ”1đ¶ ⥠âąđđ¶1đŽ = đ. (1)
Figure 1.
From the relation (1), we obtain that the
quadrilateral đđ”1đŽ1đ¶ is inscriptible and, therefore:
âąđŽ1đ”đ¶ ⥠âąđŽ1đđ¶. (2).
Also from (1), we have that đđ”1đŽđ¶1 is
inscriptible, and so
âąđŽđ”1đ¶1 ⥠âąđŽđđ¶1. (3)
Ion Patrascu, Florentin Smarandache
92
The quadrilateral MABC is inscribed, hence:
âąđđŽđ¶1 ⥠âąđ”đ¶đ. (4)
On the other hand,
âąđŽ1đđ¶ = 1800 â (đ”đ¶ïżœïżœ + đ),
âąđŽđđ¶1 = 1800 â (đđŽđ¶1 + đ).
The relation (4) drives us, together with the
above relations, to:
âąđŽ1đđ¶ ⥠âąđŽđđ¶1. (5)
Finally, using the relations (5), (2) and (3), we
conclude that: âąđŽ1đ”1đ¶ ⥠đŽđ”1đ¶1 , which justifies the
collinearity of the points đŽ1, đ”1, đ¶1.
Remark.
The Simsonâs Line is obtained in the case when
đ = 900.
2nd Lemma.
If đ is a point on the circle circumscribed to the
triangle đŽđ”đ¶ and đŽ1, đ”1, đ¶1 are points on đ”đ¶ , đ¶đŽ and
đŽđ” , respectively, such that âąđđŽ1đ¶ = âąđđ”1đ¶ =
âąđđ¶1đŽ = đ, and đđŽ1 intersects the circle a second
time in đŽâ, then đŽđŽâČ â„ đŽ1đ”1.
Proof.
The quadrilateral đđ”1đŽ1đ¶ is inscriptible (see
Figure 1); it follows that:
Complements to Classic Topics of Circles Geometry
93
âąđ¶đđŽâČ âĄ âąđŽ1đ”1đ¶. (6)
On the other hand, the quadrilateral đđŽđŽâČđ¶ is
also inscriptible, hence:
âąđ¶đđŽâČ âĄ âąđŽâČđŽđ¶. (7)
The relations (6) and (7) imply: âąđŽâČđđ¶ ⥠âąđŽâČđŽđ¶,
which gives đŽđŽâČ â„ đŽ1đ”1.
3rd Lemma. (The construction of a parallel with a given diameter
using a ruler)
In a circle of given center, construct, using only
the ruler, a parallel taken through a point of the circle
at a given diameter.
Solution.
In the given circle đ(đ, đ ) , let be a diameter
(đŽđ”)] and let đ â đ(đ, đ ). We construct the line đ”đ
(see Figure 2). We consider on this line the point đ· (đ
between đ· and đ”). We join đ· with đ , đŽ with đ and
denote đ·đ â© đŽđ = {đ}.
We take đ”đ and let {đ} = đ·đŽ â© đ”đ. The line đđ
is parallel to đŽđ”.
Constructionâs Proof.
In the triangle đ·đŽđ”, the cevians đ·đ, đŽđ and đ”đ
are concurrent.
Cevaâs Theorem provides:
Ion Patrascu, Florentin Smarandache
94
đđŽ
đđ”âđđ”
đđ·âđđ·
đđŽ= 1. (8)
But đ·đ is a median, đ·đ = đ”đ = đ .
From (8), we get đđ”
đđ·=
đđŽ
đđ· , which, by Thales
reciprocal, gives đđ â„ đŽđ”.
Figure 2.
Remark.
If we have a circle with given center and a
certain line đ, we can construct though a given point
đ a parallel to that line in such way: we take two
diameters [đ đ] and [đđ] through the center of the
given circle (see Figure 3).
Complements to Classic Topics of Circles Geometry
95
Figure 3.
We denote đ đ â© đ = {đ}; because [đ đ] ⥠[đđ], we
can construct, applying the 3rd Lemma, the parallels
through đ and đ to đ đ which intersect đ in đŸ and đż ,
respectively. Since we have on the line đ the points
đŸ, đ, đż , such that [đŸđ] ⥠[đđż] , we can construct the
parallel through đ to đ based on the construction
from 3rd Lemma.
1st Theorem. (P. Aubert â 1899)
If, through the vertices of the triangle đŽđ”đ¶, we
take three lines parallel to each other, which intersect
the circumscribed circle in đŽâČ , đ”âČ and đ¶âČ , and đ is a
Ion Patrascu, Florentin Smarandache
96
point on the circumscribed circle, as well đđŽâČ â© đ”đ¶ =
{đŽ1} , đđ”âČ â© đ¶đŽ = {đ”1} , đđ¶âČ â© đŽđ” = {đ¶1}, then đŽ1, đ”1, đ¶1
are collinear and their line is parallel to đŽđŽâČ.
Proof.
The point of the proof is to show that đđŽ1, đđ”1,
đđ¶1 generate congruent angles with đ”đ¶ , đ¶đŽ and đŽđ” ,
respectively.
đ(đđŽ1ïżœïżœ) =1
2[đ(đïżœïżœ) + đ(đ”đŽâČ )] (9)
đ(đđ”1ïżœïżœ) =1
2[đ(đïżœïżœ) + đ(đŽđ”âČ )] (10)
But đŽđŽâČ â„ đ”đ”âČ implies đ(đ”đŽâČ ) = đ(đŽđ”âČ ) , hence,
from (9) and (10), it follows that:
âąđđŽ1đ¶ ⥠âąđđ”1đ¶, (11)
đ(đđ¶1ïżœïżœ) =1
2[đ(đ”ïżœïżœ) â đ(đŽđ¶âČ )]. (12)
But đŽđŽâČ â„ đ¶đ¶âČ implies that đ(đŽđ¶âČ ) = đ(đŽâČïżœïżœ); by
returning to (12), we have that:
đ(đđ¶1ïżœïżœ) =1
2[đ(đ”ïżœïżœ) â đ(đŽđ¶âČ )] =
=1
2[đ(đ”đŽâČ ) + đ(đïżœïżœ)]. (13)
The relations (9) and (13) show that:
âąđđŽ1đ¶ ⥠âąđđ¶1đŽ. (14)
From (11) and (14), we obtain: âąđđŽ1đ¶ âĄ
âąđđ”1đ¶ ⥠âąđđ¶1đŽ , which, by 1st Lemma, verifies the
collinearity of points đŽ1, đ”1, đ¶1. Now, applying the 2nd
Lemma, we obtain the parallelism of lines đŽđŽâČ and
đŽ1đ”1.
Complements to Classic Topics of Circles Geometry
97
Figure 4.
2nd Theorem. (MâKensie â 1887)
If đŽ1đ”1đ¶1 is a transversal line in the triangle đŽđ”đ¶
(đŽ1 â đ”đ¶, đ”1 â đ¶đŽ, đ¶1 â đŽđ”), and through the triangleâs
vertices we take the chords đŽđŽâČ , đ”đ”âČ, đ¶đ¶âČ of a circle
circumscribed to the triangle, parallels with the
transversal line, then the lines đŽđŽâČ , đ”đ”âČ, đ¶đ¶âČ are
concurrent on the circumscribed circle.
Ion Patrascu, Florentin Smarandache
98
Proof.
We denote by đ the intersection of the line đŽ1đŽâČ
with the circumscribed circle (see Figure 5) and with
đ”1âČ , respectively đ¶1
âČ the intersection of the line đđ”âČ
with đŽđ¶ and of the line đđ¶âČ with đŽđ”.
Figure 5.
According to the P. Aubertâs theorem, we have
that the points đŽ1, đ”1âČ , đ¶1
âČ are collinear and that the line
đŽ1đ”1âČ is parallel to đŽđŽâČ.
From hypothesis, we have that đŽ1đ”1 â„ đŽđŽâČ; from
the uniqueness of the parallel taken through đŽ1 to đŽđŽâČ,
it follows that đŽ1đ”1 ⥠đŽ1đ”1âČ , therefore đ”1
âČ = đ”1 , and
analogously đ¶1âČ = đ¶1.
Complements to Classic Topics of Circles Geometry
99
Remark.
We have that: đđŽ1, đđ”1, đđ¶1 generate congruent
angles with đ”đ¶, đ¶đŽ and đŽđ”, respectively.
3rd Theorem. (Beltrami â 1862)
If three parallels are taken through the three
vertices of a given triangle, then their isogonals
intersect each other on the circle circumscribed to the
triangle, and vice versa.
Proof.
Let đŽđŽâČ, đ”đ”âČ, đ¶đ¶âČ the three parallel lines with a
certain direction (see Figure 6).
Ion Patrascu, Florentin Smarandache
100
Figure 6.
To construct the isogonal of the cevian đŽđŽâČ, we
take đŽâČđ â„ đ”đ¶, đ belonging to the circle circumscribed
to the triangle, having đ”đŽâČ âĄ đ¶ïżœïżœ, it follows that đŽđ
will be the isogonal of the cevian đŽđŽâČ. (Indeed, from
đ”đŽâČ âĄ đ¶ïżœïżœ it follows that âąđ”đŽđŽâČ âĄ âąđ¶đŽđ.)
On the other hand, đ”đ”âČ â„ A đŽâČ implies đ”đŽâČ âĄ
đŽïżœïżœâČ, and since đ”đŽâČ âĄ đ¶ïżœïżœ we have that đŽđ”âČ âĄ đ¶ïżœïżœ ,
which shows that the isogonal of the parallel đ”đ”âČ is
đ”đ. From đ¶đ¶âČ â„ đŽđŽâČ, it follows that đŽâČđ¶ ⥠đŽđ¶âČ, having
âąđ”âČđ¶đ ⥠âąđŽđ¶đ¶âČ, therefore the isogonal of the parallel
đ¶đ¶âČ is đ¶đâČ.
Reciprocally.
If đŽđ,đ”đ, đ¶đ are concurrent cevians in đ , the
point on the circle circumscribed to the triangle đŽđ”đ¶,
let us prove that their isogonals are parallel lines. To
construct an isogonal of đŽđ , we take đđŽâČ â„ đ”đ¶ , đŽâČ
belonging to the circumscribed circle. We have đïżœïżœ âĄ
đ”đŽâČ . Constructing the isogonal đ”đ”âČ of đ”đ, with đ”âČ on
the circumscribed circle, we will have đ¶ïżœïżœ ⥠đŽđ”âČ , it
follows that đ”đŽâČ âĄ đŽđ”âČ and, consequently, âąđŽđ”đ”âČ âĄ
âąđ”đŽđŽâČ, which shows that đŽđŽâČ â„ đ”đ”âČ. Analogously, we
show that đ¶đ¶âČ â„ đŽđŽâČ.
We are now able to solve the proposed problems.
Complements to Classic Topics of Circles Geometry
101
Solution to the 1st problem.
Using the 3rd Lemma, we construct the parallels
đŽđŽâČ, đ”đ”âČ, đ¶đ¶âČ with a certain directions of a diameter of
the circle circumscribed to the given triangle.
We join đ with đŽâČ , đ”âČ, đ¶âČ and denote the
intersection between đđŽâČ and đ”đ¶, đŽ1; đđ”âČ â© đ¶đŽ = {đ”1}
and đđŽâČ â© đŽđ = {đ¶1}.
According to the Aubertâs Theorem, the points
đŽ1, đ”1, đ¶1 will be collinear, and đđŽâČ, đđ”âČ, đđ¶âČ generate
congruent angles with đ”đ¶, đ¶đŽ and đŽđ”, respectively.
Solution to the 2nd problem.
Using the 3rd Lemma and the remark that follows
it, we construct through đŽ, đ”, đ¶ the parallels to đŽ1đ”1;
we denote by đŽâČ, đ”âČ, đ¶âČ their intersections with the
circle circumscribed to the triangle đŽđ”đ¶. (It is enough
to build a single parallel to the transversal line đŽ1đ”1đ¶1,
for example đŽđŽâČ).
We join đŽâČ with đŽ1 and denote by đ the
intersection with the circle. The point đ will be the
point we searched for. The constructionâs proof
follows from the MâKensie Theorem.
Ion Patrascu, Florentin Smarandache
102
Solution to the 3rd problem.
We suppose that đŽâČ belongs to the little arc
determined by the chord đ”ïżœïżœ in the circle
circumscribed to the triangle đŽđ”đ¶.
In this case, in order to find the isogonal đŽđŽ1, we
construct (by help of the 3rd Lemma and of the remark
that follows it) the parallel đŽâČđŽ1 to đ”đ¶, đŽ1 being on the
circumscribed circle, it is obvious that đŽđŽâČ and đŽđŽ1
will be isogonal cevians.
We suppose that đŽâČ belongs to the high arc
determined by the chord đ”ïżœïżœ; we consider đŽâČ â đŽïżœïżœ (the
arc đŽïżœïżœ does not contain the point đ¶). In this situation,
we firstly construct the parallel đ”đ to đŽđŽâČ, đ belongs
to the circumscribed circle, and then through đ we
construct the parallel đđŽ1 to đŽđ¶ , đŽ1 belongs to the
circumscribed circle. The isogonal of the line đŽđŽâČ will
be đŽđŽ1 . The constructionâs proof follows from 3rd
Lemma and from the proof of Beltramiâs Theorem.
References.
[1] F. G. M.: Exercices de GĂ©ometrie. VIII-e ed., Paris,
VI-e Librairie Vuibert, Rue de Vaugirard,77. [2] T. Lalesco: La Géométrie du Triangle. 13-e ed.,
Bucarest, 1937; Paris, Librairie Vuibert, Bd.
Saint Germain, 63. [3] C. Mihalescu: Geometria elementelor remarcabile
[The Geometry of remarkable elements]. Bucharest: Editura TehnicÄ, 1957.
Complements to Classic Topics of Circles Geometry
103
Apolloniusâs Circles
of kth Rank
The purpose of this article is to introduce
the notion of Apolloniusâs circle of k th rank.
1st Definition.
It is called an internal cevian of kth rank the line
đŽđŽđ where đŽđ â (đ”đ¶), such that đ”đŽ
đŽđđ¶= (
đŽđ”
đŽđ¶)đ (đ â â).
If đŽđâČ is the harmonic conjugate of the point đŽđ in
relation to đ” and đ¶, we call the line đŽđŽđâČ an external
cevian of kth rank.
2nd Definition.
We call Apolloniusâs circle of kth rank with
respect to the side đ”đ¶ of đŽđ”đ¶ triangle the circle which
has as diameter the segment line đŽđđŽđâČ .
1st Theorem.
Apolloniusâs circle of kth rank is the locus of
points đ from đŽđ”đ¶ triangle's plan, satisfying the
relation: đđ”
đđ¶= (
đŽđ”
đŽđ¶)đ.
Ion Patrascu, Florentin Smarandache
104
Proof.
Let đđŽđ the center of the Apolloniusâs circle of kth
rank relative to the side đ”đ¶ of đŽđ”đ¶ triangle (see
Figure 1) and đ, đ the points of intersection of this
circle with the circle circumscribed to the triangle đŽđ”đ¶.
We denote by đ· the middle of arc đ”đ¶, and we extend
đ·đŽđ to intersect the circle circumscribed in đâČ.
In đ”đâČđ¶ triangle, đâČđ· is bisector; it follows that
đ”đŽđ
đŽđđ¶=
đâČđ”
đâČđ¶= (
đŽđ”
đŽđ¶)đ, so đâČ belongs to the locus.
The perpendicular in đâČ on đâČđŽđ intersects BC on
đŽđâČâČ , which is the foot of the đ”đđ¶ triangle's outer
bisector, so the harmonic conjugate of đŽđ in relation
to đ” and đ¶, thus đŽđâČâČ = đŽđ
âČ .
Therefore, đâČ is on the Apolloniusâs circle of rank
đ relative to the side đ”đ¶, hence đâČ = đ.
Figure 3
Complements to Classic Topics of Circles Geometry
105
Let đ a point that satisfies the relation from the
statement; thus đđ”
đđ¶=
đ”đŽđ
đŽđđ¶ ; it follows â by using the
reciprocal of bisector's theorem â that đđŽđ is the
internal bisector of angle đ”đđ¶. Now let us proceed as
before, taking the external bisector; it follows that đ
belongs to the Apolloniusâs circle of center đđŽđ. We
consider now a point đ on this circle, and we
construct đ¶âČ such that âąđ”đđŽđ ⥠âąđŽđđđ¶âČ (thus (đđŽđ is
the internal bisector of the angle đ”đđ¶âČ ). Because
đŽđâČ đ â„ đđŽđ, it follows that đŽđ and đŽđ
âČ are harmonically
conjugated with respect to đ” and đ¶âČ . On the other
hand, the same points are harmonically conjugated
with respect to đ” and đ¶; from here, it follows that đ¶âČ =
đ¶, and we have đđ”
đđ¶=
đ”đŽđ
đŽđđ¶= (
đŽđ”
đŽđ¶)đ.
3rd Definition.
It is called a complete quadrilateral the
geometric figure obtained from a convex quadrilateral
by extending the opposite sides until they intersect. A
complete quadrilateral has 6 vertices, 4 sides and 3
diagonals.
2nd Theorem.
In a complete quadrilateral, the three diagonals'
middles are collinear (Gauss - 1810).
Ion Patrascu, Florentin Smarandache
106
Proof.
Let đŽđ”đ¶đ·đžđč a given complete quadrilateral (see
Figure 2). We denote by đ»1, đ»2, đ»3, đ»4 respectively the
orthocenters of đŽđ”đč, đŽđ·đž, đ¶đ”đž, đ¶đ·đč triangles, and
let đŽ1, đ”1, đč1 the feet of the heights of đŽđ”đč triangle.
Figure 4
As previously shown, the following relations
occur: đ»1đŽ.đ»1đŽ1 â đ»1đ”.đ»1đ”1 = đ»1đč.đ»1đč1; they express
that the point đ»1 has equal powers to the circles of
diameters đŽđ¶, đ”đ·, đžđč , because those circles contain
respectively the points đŽ1, đ”1, đč1, and đ»1 is an internal
point.
It is shown analogously that the points đ»2, đ»3, đ»4
have equal powers to the same circles, so those points
are situated on the radical axis (common to the
circles), therefore the circles are part of a fascicle, as
Complements to Classic Topics of Circles Geometry
107
such their centers â which are the middles of the
complete quadrilateral's diagonals â are collinear.
The line formed by the middles of a complete
quadrilateral's diagonals is called Gaussâs line or
Gauss-Newtonâs line.
3rd Theorem.
The Apolloniusâs circle of kth rank of a triangle
are part of a fascicle.
Proof.
Let đŽđŽđ , đ”đ”đ , đ¶đ¶đ be concurrent cevians of kth
rank and đŽđŽđâČ , đ”đ”đ
âČ , đ¶đ¶đâČ be the external cevians of kth
rank (see Figure 3). The figure đ”đâČ đ¶đđ”đđ¶đ
âČđŽđđŽđâČ is a
complete quadrilateral and 2nd theorem is applied.
Figure 5
Ion Patrascu, Florentin Smarandache
108
4th Theorem.
The Apolloniusâs circle of kth rank of a triangle
are the orthogonals of the circle circumscribed to the
triangle.
Proof.
We unite đ to đ· and đ (see Figure 1), đđ· â„ đ”đ¶
and đ(đŽđđđŽđâČ ) = 900, it follows that đđŽđ
âČ đŽđ = đđ·đŽïżœïżœ =
đđđŽïżœïżœ.
The congruence đđŽđâČ đŽđ
⥠đđđŽïżœïżœ shows that đđ
is tangent to the Apolloniusâs circle of center đđŽđ.
Analogously, it can be demonstrated for the
other Apolloniusâs Circle.
1st Remark.
The previous theorem indicates that the radical
axis of Apolloniusâs circle of kth rank is the perpen-
dicular taken from đ to the line đđŽđđđ”đ
.
5th Theorem.
The centers of Apolloniusâs Circle of kth rank of a
triangle are situated on the trilinear polar associated
to the intersection point of the cevians of 2kth rank.
Complements to Classic Topics of Circles Geometry
109
Proof.
From the previous theorem, it results that đđ â„
đđđŽđ, so đđđŽđ
is an external cevian of rank 2 for đ”đ¶đ
triangle, thus an external symmedian. Henceforth, đđŽđ
đ”
đđŽđđ¶
= (đ”đ
đ¶đ)2
= (đŽđ”
đŽđ¶)2đ
(the last equality occurs because
đ belong to the Apolloniusâs circle of rank đ associated
to the vertex đŽ).
6th Theorem.
The Apolloniusâs circle of kth rank of a triangle
intersects the circle circumscribed to the triangle in
two points that belong to the internal and external
cevians of k+1th rank.
Proof.
Let đ and đ points of intersection of the
Apolloniusâs circle of center đđŽđ with the circle
circumscribed to the đŽđ”đ¶ (see Figure 1). We take from
đ and đ the perpendiculars đđ1 , đđ2 and đđ1, đđ2 on
đŽđ” and đŽđ¶ respectively. The quadrilaterals đŽđ”đđ¶ ,
đŽđ”đ¶đ are inscribed, it follows the similarity of
triangles đ”đđ1, đ¶đđ2 and đ”đđ1, đ¶đđ2, from where we
get the relations: đ”đ
đ¶đ=
đđ1
đđ2,
đđ”
đđ¶=
đđ1
đđ2.
Ion Patrascu, Florentin Smarandache
110
But đ”đ
đ¶đ= (
đŽđ”
đŽđ¶)đ,
đđ”
đđ¶= (
đŽđ”
đŽđ¶)đ,
đđ1
đđ2= (
đŽđ”
đŽđ¶)đ and
đđ1
đđ2=
(đŽđ”
đŽđ¶)đ
, relations that show that đ and đ belong
respectively to the internal cevian and the external
cevian of rank đ + 1.
4th Definition.
If the Apolloniusâs circle of kth rank associated
with a triangle has two common points, then we call
these points isodynamic points of kth rank (and we
denote them đđ ,đđâČ).
1st Property.
If đđ ,đđâČ are isodynamic centers of kth rank,
then:
đđđŽ. đ”đ¶đ = đđđ”. đŽđ¶đ = đđđ¶. đŽđ”đ;
đđâČđŽ. đ”đ¶đ = đđ
âČđ”. đŽđ¶đ = đđâČđ¶. đŽđ”đ.
The proof of this property follows immediately
from 1st Theorem.
2nd Remark.
The Apolloniusâs circle of 1st rank is the
investigated Apolloniusâs circle (the bisectors are
cevians of 1st rank). If đ = 2, the internal cevians of 2nd
rank are the symmedians, and the external cevians of
2nd rank are the external symmedians, i.e. the tangents
Complements to Classic Topics of Circles Geometry
111
in triangleâs vertices to the circumscribed circle. In
this case, for the Apolloniusâs circle of 2nd rank, the 3rd
Theorem becomes:
7th Theorem.
The Apolloniusâs circle of 2nd rank intersects the
circumscribed circle to the triangle in two points
belonging respectively to the antibisector's isogonal
and to the cevian outside of it.
Proof.
It follows from the proof of the 6th theorem. We
mention that the antibisector is isotomic to the
bisector, and a cevian of 3rd rank is isogonic to the
antibisector.
Ion Patrascu, Florentin Smarandache
112
References.
[1] N. N. MihÄileanu: LecÈii complementare de geometrie [Complementary Lessons of Geometry], Editura DidacticÄ Èi PedagogicÄ,
BucureÈti, 1976. [2] C. Mihalescu: Geometria elementelor remarcabile
[The Geometry of Outstanding Elements],
Editura TehnicÄ, BucureÈti, 1957. [3] V. Gh. VodÄ: Triunghiul â ringul cu trei colÈuri [The
Triangle-The Ring with Three Corners], Editura Albatros, BucureÈti, 1979.
[4] F. Smarandache, I. PÄtraÈcu: Geometry of
Homological Triangle, The Education Publisher Inc., Columbus, Ohio, SUA, 2012.
Complements to Classic Topics of Circles Geometry
113
Apolloniusâs Circle of Second
Rank
This article highlights some properties of
Apolloniusâs circle of second rank in
connection with the adjoint circles and the
second Brocardâs triangle.
1st Definition.
It is called Apolloniusâs circle of second rank
relative to the vertex đŽ of the triangle đŽđ”đ¶ the circle
constructed on the segment determined on the
simediansâ feet from đŽ on đ”đ¶ as diameter.
1st Theorem.
The Apolloniusâs circle of second rank relative to
the vertex đŽ of the triangle đŽđ”đ¶ intersect the
circumscribed circle of the triangle đŽđ”đ¶ in two points
belonging respectively to the cevian of third rank
(antibisectorâs isogonal) and to its external cevian.
The theoremâs proof follows from the theorem
relative to the Apolloniusâs circle of kth rank (see [1]).
Ion Patrascu, Florentin Smarandache
114
1st Proposition.
The Apolloniusâs circle of second rank relative to
the vertex đŽ of the triangle đŽđ”đ¶ intersects the
circumscribed circle in two points đ and đ (đ on the
same side of đ”đ¶ as đŽ). Then, (đđ is a bisector in the
triangle đđ”đ¶ , S is the simedianâs foot from đŽ of the
triangle đŽđ”đ¶.
Proof.
đ belongs to the Apolloniusâs circle of second
rank, therefore:
đđ”
đđ¶= (
đŽđ”
đŽđ¶)2. (1)
Figure 1.
On the other hand, đ being the simedianâs foot,
we have:
Complements to Classic Topics of Circles Geometry
115
đđ”
đđ¶= (
đŽđ”
đŽđ¶)2. (2)
From relations (1) and (2), we note that đđ”
đđ¶=
đđ”
đđ¶ ,
a relation showing that đđ is bisector in the triangle
đđ”đ¶.
Remarks.
1. The Apolloniusâs circle of second relative
to the vertex đŽ of the triangle đŽđ”đ¶ (see
Figure 1) is an Apolloniusâs circle for the
triangle đđ”đ¶. Indeed, we proved that đđ
is an internal bisector in the triangle đđ”đ¶,
and since đâČ, the external simedianâs foot
of the triangle đŽđ”đ¶ , belongs to the
Apolloniusâs Circle of second rank, we
have đ(âąđâČđđ) = 900 , therefore đđâ is an
external bisector in the triangle đđ”đ¶.
2. đđ is a simedian in đđ”đ¶ . Indeed, the
Apolloniusâs circle of second rank, being
an Apolloniusâs circle for đđ”đ¶, intersects
the circle circum-scribed to đđ”đ¶ after đđ,
which is simedian in this triangle.
2nd Definition.
It is called adjoint circle of a triangle the circle
that passes through two vertices of the triangle and in
Ion Patrascu, Florentin Smarandache
116
one of them is tangent to the triangleâs side. We denote
(đ”ïżœïżœ) the adjoint circle that passes through đ” and đŽ,
and is tangent to the side đŽđ¶ in đŽ.
About the circles (đ”ïżœïżœ) and (đ¶ïżœïżœ), we say that they
are adjoint to the vertex đŽ of the triangle đŽđ”đ¶.
3rd Definition.
It is called the second Brocardâs triangle the
triangle đŽ2đ”2đ¶2 whose vertices are the projections of
the center of the circle circumscribed to the triangle
đŽđ”đ¶ on triangleâs simedians.
2nd Proposition.
The Apolloniusâs circle of second rank relative to
the vertex đŽ of triangle đŽđ”đ¶ and the adjoint circles
relative to the same vertex đŽ intersect in vertex đŽ2 of
the second Brocardâs triangle.
Proof.
It is known that the adjoint circles (đ”ïżœïżœ) and (đ¶ïżœïżœ)
intersect in a point belonging to the simedian đŽđ; we
denote this point đŽ2 (see [3]).
We have:
âąđ”đŽ2đ = âąđŽ2đ”đŽ + âąđŽ2đŽđ”,
but:
âąđŽ2đ”đŽ ⥠âąđ”đŽ2đ = âąđŽ2đŽđ” + đŽ2đŽđ¶ = âąđŽ.
Complements to Classic Topics of Circles Geometry
117
Analogously, âąđ¶đŽ2đ = âąđŽ , therefore (đŽ2đ is the
bisector of the angle đ”đŽ2đ¶. The bisectorâs theorem in
this triangle leads to: đđ”
đđ¶=
đ”đŽ2
đ¶đŽ2 ,
but:
đđ”
đđ¶= (
đŽđ”
đŽđ¶)2,
consequently:
đ”đŽ2
đ¶đŽ2= (
đŽđ”
đŽđ¶)2,
so đŽ2 is a point that belongs to the Apolloniusâs circle
of second rank.
Figure 2.
We prove that đŽ2 is a vertex in the second
Brocardâs triangle, i.e. đđŽ2 â„ đŽđ, O the center of the
circle circumscribed to the triangle đŽđ”đ¶.
Ion Patrascu, Florentin Smarandache
118
We pointed (see Figure 2) that đ(đ”đŽ2ïżœïżœ) = 2đŽ, if
âąđŽ is an acute angle, then also đ(đ”đïżœïżœ) = 2đŽ ,
therefore the quadrilateral đđ¶đŽ2đ” is inscriptible.
Because đ(đđ¶ïżœïżœ) = 900 â đ(ïżœïżœ) , it follows that
đ(đ”đŽ2ïżœïżœ) = 900 + đ(ïżœïżœ).
On the other hand, đ(đŽđŽ2ïżœïżœ) = 1800 â đ(ïżœïżœ), so
đ(đ”đŽ2ïżœïżœ) + đ(đŽđŽ2ïżœïżœ) = 2700 and, consequently, đđŽ2 â„
đŽđ.
Remarks.
1. If đ(ïżœïżœ) < 900 , then four remarkable
circles pass through đŽ2 : the two circles
adjoint to the vertex đŽ of the triangle đŽđ”đ¶,
the circle circumscribed to the triangle
đ”đđ¶ (where đ is the center of the
circumscribed circle) and the Apolloniusâs
circle of second rank corresponding to the
vertex đŽ.
2. The vertex đŽ2 of the second Brocardâs
triangle is the middle of the chord of the
circle circumscribed to the triangle đŽđ”đ¶
containing the simedian đŽđ.
3. The points đ , đŽ2 and đâ (the foot of the
external simedian to đŽđ”đ¶) are collinear.
Indeed, we proved that đđŽ2 â„ đŽđ; on the
other hand, we proved that (đŽ2đ is an
internal bisector in the triangle đ”đŽ2đ¶, and
Complements to Classic Topics of Circles Geometry
119
since đâČđŽ2 â„ đŽđ , the outlined collinearity
follows from the uniqueness of the
perpendicular in đŽ2 on đŽđ.
Open Problem.
The Apolloniusâs circle of second rank relative to
the vertex đŽ of the triangle đŽđ”đ¶ intersects the circle
circumscribed to the triangle đŽđ”đ¶ in two points đ and
đ (đ and đŽ apart of đ”đ¶).
We denote by đ the second point of intersection
between the line đŽđ and the Apolloniusâs circle of
second rank.
What can we say about đ?
Is đ a remarkable point in triangleâs geometry?
Ion Patrascu, Florentin Smarandache
120
References.
[1] I. Patrascu, F. Smarandache: Cercurile Apollonius de rangul k [The Apolloniusâs Circle of kth rank]. In: âRecreaÈii matematiceâ, Anul XVIII, nr.
1/2016, IaÈi, RomĂąnia, p. 20-23. [2] I. Patrascu: Axe Èi centre radicale ale cercurilor
adjuncte ale unui triunghi [Axes and Radical
Centers of Adjoint Circles of a Triangle]. In: âRecreaÈii matematiceâ, Anul XVII, nr. 1/2010,
IaÈi, RomĂąnia. [3] R. Johnson: Advanced Euclidean Geometry. New
York: Dover Publications, Inc. Mineola, 2007.
[4] I. Patrascu, F. Smarandache: Variance on topics of plane geometry. Columbus: The Educational Publisher, Ohio, USA, 2013.
Complements to Classic Topics of Circles Geometry
121
A Sufficient Condition for
the Circle of the 6 Points
to Become Eulerâs Circle
In this article, we prove the theorem
relative to the circle of the 6 points and,
requiring on this circle to have three other
remarkable triangleâs points, we obtain the
circle of 9 points (the Eulerâs C ircle).
1st Definition.
It is called cevian of a triangle the line that
passes through the vertex of a triangle and an opposite
sideâs point, other than the two left vertices.
Figure 1.
Ion Patrascu, Florentin Smarandache
122
1st Remark.
The name cevian was given in honor of the
italian geometrician Giovanni Ceva (1647 - 1734).
2nd Definition.
Two cevians of the same triangleâs vertex are
called isogonal cevians if they create congruent angles
with triangleâs bisector taken through the same vertex.
2nd Remark.
In the Figure 1 we represented the bisector đŽđ·
and the isogonal cevians đŽđŽ1 and đŽđŽ2. The following
relations take place:
đŽ1đŽïżœïżœ ⥠đŽ2đŽïżœïżœ;
đ”đŽđŽ1 ⥠đ¶đŽđŽ2.
1st Proposition.
In a triangle, the height and the radius of the
circumscribed circle corresponding to a vertex are
isogonal cevians.
Proof.
Let đŽđ”đ¶ an acute-angled triangle with the height
đŽđŽâČ and the radius đŽđ (see Figure 2).
Complements to Classic Topics of Circles Geometry
123
Figure 2.
The angle đŽđđ¶ is a central angle, and the angle
đŽđ”đ¶ is an inscribed angle, so đŽđïżœïżœ = 2đŽđ”ïżœïżœ. It follows
that đŽđïżœïżœ = 900 â ïżœïżœ.
On the other hand, đ”đŽđŽâČ = 900 â ïżœïżœ , so đŽđŽâČ and
đŽđ are isogonal cevians.
The theorem can be analogously proved for the
obtuse triangle.
3rd Remark.
One can easily prove that in a triangle, if đŽđ is
circumscribed circleâs radius, its isogonal cevian is the
triangleâs height from vertex đŽ.
3rd Definition.
Two points đ1, đ2 in the plane of triangle đŽđ”đ¶ are
called isogonals (isogonal conjugated) if the ceviansâ
pairs (đŽđ1, đŽđ2), (đ”đ1, đ”đ2), (đ¶đ1, đ¶đ2), are composed
of isogonal cevians.
Ion Patrascu, Florentin Smarandache
124
4th Remark.
In a triangle, the orthocenter and circumscribed
circleâs center are isogonal points.
1st Theorem. (The 6 points circle)
If đ1 and đ2 are two isogonal points in the
triangle đŽđ”đ¶ , and đŽ1, đ”1, đ¶1 respectively đŽ2, đ”2, đ¶2 are
their projections on the sides đ”đ¶ , đ¶đŽ and đŽđ” of the
triangle, then the points đŽ1, đŽ2, đ”1, đ”2, đ¶1, đ¶2 are
concyclical.
Proof.
The mediator of segment [đŽ1đŽ2] passes through
the middle đ of segment [đ1, đ2] because the trapezoid
đ1đŽ1đŽ2đ2 is rectangular and the mediator of [đŽ1đŽ2]
contains its middle line, therefore (see Figure 3), we
have: đđŽ1 = đđŽ2 (1). Analogously, it follows that the
mediators of segments [đ”1đ”2] and [đ¶1đ¶2] pass through
đ, so đđ”1 = đđ”2 (2) and đđ¶1 = đđ¶2 (3). We denote by
đŽ3 and đŽ4 respectively the middles of segments [đŽđ1]
and [đŽđ2 ]. We prove that the triangles đđŽ3đ¶1 and
đ”2đŽ4đ are congruent. Indeed, đđŽ3 =1
2đŽđ2 (middle
line), and đ”2đŽ4 =1
2đŽđ2, because it is a median in the
rectangular triangle đ2đ”2đŽ , so đđŽ3 = đ”2đŽ4 ;
analogously, we obtain that đŽ4đ = đŽ3đ¶1 .
Complements to Classic Topics of Circles Geometry
125
Figure 3.
We have that:
đđŽ3đ¶1 = đđŽ3đ1
+ đ1đŽ3đ¶1 = đ1đŽđ2 + 2đ1đŽđ¶1 =
= ïżœïżœ + đ1đŽïżœïżœ;
đ”2đŽ4ïżœïżœ = đ”2đŽ4đ2 + đđŽ4đ2
= đ1đŽđ2 + 2đ2đŽđ”2 =
= ïżœïżœ + đ2đŽïżœïżœ.
But đ1đŽïżœïżœ = đ2đŽïżœïżœ , because the cevians đŽđ1 and
đŽđ2 are isogonal and therefore đđŽ3đ¶1 = đ”2đŽ4ïżœïżœ. Since
âđđŽ3đ¶1 = âđ”2đŽ4đ, it follows that đđ”2 = đđ¶1 (4).
Repeating the previous reasoning for triangles
đđ”3đ¶1 and đŽ2đ”4đ , where đ”3 and đ”4 are respectively
the middles of segments (đ”đ1) and (đ”đ2), we find that
they are congruent and it follows that đđ¶1 = đđŽ2 (5).
The relations (1) - (5) lead to đđŽ1 = đđŽ2 = đđ”1 =
đđ”2 = đđ¶1 = đđ¶2 , which shows that the points
đŽ1, đŽ2, đ”1, đ”2, đ¶1, đ¶2 are located on a circle centered in đ,
Ion Patrascu, Florentin Smarandache
126
the middle of the segment determined by isogonal
points given by đ1 and đ2.
4th Definition.
It is called the 9 points circle or Eulerâs circle of
the triangle đŽđ”đ¶ the circle that contains the middles
of triangleâs sides, the triangle heightsâ feet and the
middles of the segments determined by the
orthocenter with triangleâs vertex.
2nd Proposition.
If đ1, đ2 are isogonal points in the triangle đŽđ”đ¶
and if on the circle of their corresponding 6 points
there also are the middles of the segments (đŽđ1), (đ”đ1),
(đ¶đ1 ), then the 6 points circle coincides with the
Eulerâs circle of the triangle đŽđ”đ¶.
1st Proof.
We keep notations from Figure 3; we proved that
the points đŽ1, đŽ2, đ”1, đ”2, đ¶1, đ¶2 are on the 6 points circle
of the triangle đŽđ”đ¶, having its center in đ, the middle
of segment [đ1đ2].
If on this circle are also situated the middles
đŽ3, đ”3, đ¶3 of segments (đŽđ1), (đ”đ1), (đ¶đ1), then we have
đđŽ3 = đđ”3 = đđ¶3.
Complements to Classic Topics of Circles Geometry
127
We determined that đđŽ3 is middle line in the
triangle đ1đ2đŽ , therefore đđŽ3 =1
2đŽđ2 , analogously
đđ”3 =1
2đ”đ2 and đđ¶3 =
1
2đ¶đ2, and we obtain that đ2đŽ =
đ2đ” = đ2đ¶, consequently đ is the center of the circle
circumscribed to the triangle đŽđ”đ¶, so đ2 = đ.
Because đ1 is the isogonal of đ, it follows that
đ1 = đ», therefore the circle of 6 points of the isogonal
points đ and đ» is the circle of 9 points.
2nd Proof.
Because đŽ3đ”3 is middle line in the triangle đ1đŽđ”,
it follows that
âąđ1đŽđ” ⥠âąđ1đŽ2đ”3. (1)
Also, đŽ3đ¶3 is middle line in the triangle đ1đŽđ¶, and
đŽ3đ¶3 is middle line in the triangle đ1đŽđ2, therefore we
get
âąđđŽ3đ¶3 ⥠âąđ2đŽđ¶. (2)
The relations (1), (2) and the fact that đŽđ1 and
đŽđ2 are isogonal cevians lead to:
âąđ1đŽ2đ”3 ⥠đđŽ3đ¶3. (3)
The point đ is the center of the circle
circumscribed to đŽ3đ”3đ¶3 ; then, from (3) and from
isogonal ceviansâ properties, one of which is
circumscribed circle radius, it follows that in the
triangle đŽ3đ”3đ¶3 the line đ1đŽ3 is a height, as đ”3đ¶3 â„ đ”đ¶,
we get that đ1đŽ is a height in the triangle ABC and,
therefore, đ1 will be the orthocenter of the triangle
Ion Patrascu, Florentin Smarandache
128
đŽđ”đ¶ , and đ2 will be the center of the circle
circumscribed to the triangle đŽđ”đ¶.
5th Remark.
Of those shown, we get that the center of the
circle of 9 points is the middle of the line determined
by triangleâs orthocenter and by the circumscribed
circleâs center, and the fact that Eulerâs circle radius is
half the radius of the circumscribed circle.
References.
[1] Roger A. Johnson: Advanced Euclidean Geometry. New York: Dover Publications, 2007.
[2] CÄtÄlin Barbu: Teoreme fundamentale din geometria triunghiului [Fundamental theorems of triangleâs geometry]. BacÄu: Editura Unique,
2008.
Complements to Classic Topics of Circles Geometry
129
An Extension of a Problem
of Fixed Point
In this article, we extend the requirement
of the Problem 9.2 proposed at Varna 2015
Spring Competition , both in terms of
membership of the measure đŸ, and the case
for the problem for the ex-inscribed circle
đ¶. We also try to guide the student in the
search and identification of the fixed point,
for succeeding in solving any problem of
this type.
The statement of the problem is as follows:
âWe fix an angle đŸ â (0, 900) and the line
đŽđ” which divides the plane in two half-planes đŸ
and ïżœïżœ. The point đ¶ in the half-plane đŸ is situated
such that đ(đŽđ¶ïżœïżœ) = đŸ. The circle inscribed in the
triangle đŽđ”đ¶ with the center đŒ is tangent to the
sides đŽđ¶ and đ”đ¶ in the points đč and đž ,
respectively. The point đ is located on the
segment line (đŒđž , the point đž between đŒ and đ
such that đđž â„ đ”đ¶ and đđž = đŽđč. The point đ is
situated on the segment line (đŒđč, such that đč is
between đŒ and đ ; đđč â„ đŽđ¶ and đđč = đ”đž . Prove
Ion Patrascu, Florentin Smarandache
130
that the mediator of segment đđ passes through
a fixed point.â (Stanislav Chobanov)
Proof.
Firstly, it is useful to note that the point đ¶ varies
in the half-plane đ on the arc capable of angle đŸ; we
know as well that đ(đŽđŒïżœïżœ) = 900 +đŸ
2, so đŒ varies on the
arc capable of angle of measure 900 +đŸ
2 situated in the
half-plane đ.
Another useful remark is about the segments đŽđč
and đ”đž , which in a triangle have the lengths đ â đ ,
respectively đ â đ , where đ is the half-perimeter of
the triangle đŽđ”đ¶ with đŽđ” = đ â constant; therefore,
we have âđđžđ” ⥠âđŽđčđ with the consequence đđ” = đđŽ.
Considering the vertex đ¶ of the triangle đŽđ”đ¶ the
middle of the arc capable of angle đŸ built on đŽđ”, we
observe that đđ is parallel to đŽđ” ; more than that,
đŽđ”đđ is an isosceles trapezoid, and segment đđ
mediator will be a symmetry axis of the trapezoid, so
it will coincide with the mediator of đŽđ”, which is a
fixed line, so we're looking for the fixed point on
mediator of đŽđ”.
Let đ· be the intersection of the mediators of
segments đđ and đŽđ” , see Figure 1, where we
considered đ(ïżœïżœ) < đ(ïżœïżœ) . The point đ· is on the
mediator of đŽđ”, so we have đ·đŽ = đ·đ”; the point đ· is
also on the mediator of đđ, so we have đ·đ = đ·đ; it
Complements to Classic Topics of Circles Geometry
131
follows that: âđđ”đ· ⥠âđđŽđ·, a relation from where we
get that âąđđŽđ· = âąđđ”đ·.
Figure 1
If we denote đ(đđŽïżœïżœ) = đ„ and đ(đ·đŽïżœïżœ) = đŠ , we
have đđŽïżœïżœ = đ„ + đŽ + đŠ , đđ”ïżœïżœ = 3600 â đ” â đŠ â (900 â đ„).
From đ„ + đŽ + đŠ = 3600 â đ” â đŠ â 900 + đ„, we find
that đŽ + đ” + 2đŠ = 2700, and since đŽ + đ” = 1800 â đŸ, we
find that 2đŠ = 900 â đŸ, therefore the requested fixed
point đ· is the vertex of triangle đ·đŽđ” , situated in ïżœïżœ
such that đ(đŽđ·ïżœïżœ) = 900 â đŸ.
Ion Patrascu, Florentin Smarandache
132
1st Remark.
If đŸ = 900, we propose to the reader to prove that
the quadrilateral đŽđ”đđ is a parallelogram; in this case,
the requested fixed point does not exist (or is the point
at infinity of the perpendicular lines to đŽđ”).
2nd Remark.
If đŸ â (900, 1800), the problem applies, and we
find that the fixed point đ· is located in the half-plane
đ , such that the triangle đ·đŽđ” is isosceles, having
đ(đŽđïżœïżœ) = đŸ â 900.
We suggest to the reader to solve the following
problem:
We fix an angle đŸ â (00, 1800) and the line
AB which divides the plane in two half-planes, đ
and ïżœïżœ. The point đ¶ in the half-plane đ is located
such that đ(đŽđ¶ïżœïżœ) = đŸ . The circle đ¶ â ex-
inscribed to the triangle đŽđ”đ¶ with center đŒđ is
tangent to the sides đŽđ¶ and đ”đ¶ in the points đč
and đž, respectively. The point đ is located on the
line segment (đŒđđž , đž is between đŒđ and đ such
that đđž â„ đ”đ¶ and đđž = đŽđč . The point đ is
located on the line segment (đŒđđč such that đč is
between đŒ and đ , đđč â„ đŽđ¶ and đđč = đ”đž . Prove
that the mediator of the segment đđ passes
through a fixed point.
Complements to Classic Topics of Circles Geometry
133
3rd Remark.
As seen, this problem is also true in the case đŸ =
900, more than that, in this case, the fixed point is the
middle of đŽđ”. Prove!
References.
[1] Ion Patrascu: Probleme de geometrie planÄ [Planar
Geometry Problems]. Craiova: Editura Cardinal, 1996.
Ion Patrascu, Florentin Smarandache
134
Complements to Classic Topics of Circles Geometry
135
Some Properties of the
Harmonic Quadrilateral
In this article, we review some properties
of the harmonic quadrilateral related to
triangle simedians and to Apolloniusâs
Circle.
1st Definition.
A convex circumscribable quadrilateral đŽđ”đ¶đ·
having the property đŽđ” â đ¶đ· = đ”đ¶ â đŽđ· is called
harmonic quadrilateral.
2nd Definition.
A triangle simedian is the isogonal cevian of a
triangle median.
1st Proposition.
In the triangle đŽđ”đ¶, the cevian đŽđŽ1, đŽ1 â (đ”đ¶) is
a simedian if and only if đ”đŽ1
đŽ1đ¶= (
đŽđ”
đŽđ¶)2. For Proof of this
property, see infra.
Ion Patrascu, Florentin Smarandache
136
Figura 1.
2nd Proposition.
In an harmonic quadrilateral, the diagonals are
simedians of the triangles determined by two
consecutive sides of a quadrilateral with its diagonal.
Proof.
Let đŽđ”đ¶đ· be an harmonic quadrilateral and
{đŸ} = đŽđ¶ â© đ”đ· (see Figure 1). We prove that đ”đŸ is
simedian in the triangle đŽđ”đ¶.
From the similarity of the triangles đŽđ”đŸ and
đ·đ¶đŸ, we find that: đŽđ”
đ·đ¶=
đŽđŸ
đ·đŸ=
đ”đŸ
đ¶đŸ . (1)
From the similarity of the triangles đ”đ¶đŸ Ći đŽđ·đŸ,
we conclude that:
Complements to Classic Topics of Circles Geometry
137
đ”đ¶
đŽđ·=
đ¶đŸ
đ·đŸ=
đ”đŸ
đŽđŸ . (2)
From the relations (1) and (2), by division, it
follows that: đŽđ”
đ”đ¶.đŽđ·
đ·đ¶=
đŽđŸ
đ¶đŸ . (3)
But đŽđ”đ¶đ· is an harmonic quadrilateral;
consequently, đŽđ”
đ”đ¶=
đŽđ·
đ·đ¶ ;
substituting this relation in (3), it follows that:
(đŽđ”
đ”đ¶)2
=đŽđŸ
đ¶đŸ;
As shown by Proposition 1, đ”đŸ is a simedian in
the triangle đŽđ”đ¶. Similarly, it can be shown that đŽđŸ is
a simedian in the triangle đŽđ”đ·, that đ¶đŸ is a simedian
in the triangle đ”đ¶đ·, and that đ·đŸ is a simedian in the
triangle đŽđ·đ¶.
Remark 1.
The converse of the 2nd Proposition is proved
similarly, i.e.:
3rd Proposition.
If in a convex circumscribable quadrilateral, a
diagonal is a simedian in the triangle formed by the
other diagonal with two consecutive sides of the
quadrilateral, then the quadrilateral is an harmonic
quadrilateral.
Ion Patrascu, Florentin Smarandache
138
Remark 2.
From 2nd and 3rd Propositions above, it results a
simple way to build an harmonic quadrilateral.
In a circle, let a triangle đŽđ”đ¶ be considered; we
construct the simedian of A, be it đŽđŸ, and we denote
by D the intersection of the simedian đŽđŸ with the
circle. The quadrilateral đŽđ”đ¶đ· is an harmonic
quadrilateral.
Proposition 4.
In a triangle đŽđ”đ¶, the points of the simedian of A
are situated at proportional lengths to the sides đŽđ”
and đŽđ¶.
Figura 2.
Complements to Classic Topics of Circles Geometry
139
Proof.
We have the simedian đŽđŽ1 in the triangle đŽđ”đ¶
(see Figure 2). We denote by đ· and đž the projections
of đŽ1 on đŽđ”, and đŽđ¶ respectively.
We get:
đ”đŽ1
đ¶đŽ1=
đŽđđđâ(đŽđ”đŽ1)
đŽđđđâ(đŽđ¶đŽ1)=
đŽđ” â đŽ1đ·
đŽđ¶ â đŽ1đž .
Moreover, from 1st Proposition, we know that
đ”đŽ1
đŽ1đ¶= (
đŽđ”
đŽđ¶)2
.
Substituting in the previous relation, we obtain
that: đŽ1đ·
đŽ1đž=
đŽđ”
đŽđ¶ .
On the other hand, đ·đŽ1 = đŽđŽ1. From đ”đŽđŽ1 and
đŽ1đž = đŽđŽ1 â đ đđđ¶đŽđŽ1, hence: đŽ1đ·
đŽ1đž=
đ đđđ”đŽđŽ1
đ đđđ¶đŽđŽ1=
đŽđ”
đŽđ¶ . (4)
If M is a point on the simedian and đđ1 and đđ2
are its projections on đŽđ” , and đŽđ¶ respectively, we
have:
đđ1 = đŽđ â đ đđđ”đŽđŽ1, đđ2 = đŽđ â đ đđđ¶đŽđŽ1,
hence:
đđ1
đđ2=
đ đđđ”đŽđŽ1
đ đđđ¶đŽđŽ1
.
Taking (4) into account, we obtain that: đđ1
đđ2=
đŽđ”
đŽđ¶ .
Ion Patrascu, Florentin Smarandache
140
3rd Remark.
The converse of the property in the statement
above is valid, meaning that, if đ is a point inside a
triangle, its distances to two sides are proportional to
the lengths of these sides. The point belongs to the
simedian of the triangle having the vertex joint to the
two sides.
5th Proposition.
In an harmonic quadrilateral, the point of
intersection of the diagonals is located towards the
sides of the quadrilateral to proportional distances to
the length of these sides. The Proof of this Proposition
relies on 2nd and 4th Propositions.
6th Proposition. (R. Tucker)
The point of intersection of the diagonals of an
harmonic quadrilateral minimizes the sum of squares
of distances from a point inside the quadrilateral to
the quadrilateral sides.
Proof.
Let đŽđ”đ¶đ· be an harmonic quadrilateral and đ
any point within.
Complements to Classic Topics of Circles Geometry
141
We denote by đ„, đŠ, đ§, đą the distances of đ to the
đŽđ” , đ”đ¶ , đ¶đ· , đ·đŽ sides of lenghts đ, đ, đ, and đ (see
Figure 3).
Figure 3.
Let đ be the đŽđ”đ¶đ· quadrilateral area.
We have:
đđ„ + đđŠ + đđ§ + đđą = 2đ.
This is true for đ„, đŠ, đ§, đą and đ, đ, đ, đ real numbers.
Following Cauchy-Buniakowski-Schwarz Ine-
quality, we get:
(đ2 + đ2 + đ2 + đ2)(đ„2 + đŠ2 + đ§2 + đą2)
â„ (đđ„ + đđŠ + đđ§ + đđą)2 ,
and it is obvious that:
đ„2 + đŠ2 + đ§2 + đą2 â„4đ2
đ2 + đ2 + đ2 + đ2 .
We note that the minimum sum of squared
distances is:
Ion Patrascu, Florentin Smarandache
142
4đ2
đ2 + đ2 + đ2 + đ2= đđđđ đĄ.
In Cauchy-Buniakowski-Schwarz Inequality, the
equality occurs if: đ„
đ=
đŠ
đ=
đ§
đ=
đą
đ .
Since {đŸ} = đŽđ¶ â© đ”đ· is the only point with this
property, it ensues that đ = đŸ, so đŸ has the property
of the minimum in the statement.
3rd Definition.
We call external simedian of đŽđ”đ¶ triangle a
cevian đŽđŽ1â corresponding to the vertex đŽ, where đŽ1â
is the harmonic conjugate of the point đŽ1 â simedianâs
foot from đŽ relative to points đ” and đ¶.
4th Remark.
In Figure 4, the cevian đŽđŽ1 is an internal
simedian, and đŽđŽ1â is an external simedian.
We have:
đŽ1đ”
đŽ1đ¶=
đŽ1âČđ”
đŽ1âČđ¶ .
In view of 1st Proposition, we get that:
đŽ1âČđ”
đŽ1âČđ¶= (
đŽđ”
đŽđ¶)2
.
Complements to Classic Topics of Circles Geometry
143
7th Proposition.
The tangents taken to the extremes of a diagonal
of a circle circumscribed to the harmonic quadrilateral
intersect on the other diagonal.
Proof.
Let đ be the intersection of a tangent taken in đ·
to the circle circumscribed to the harmonic
quadrilateral đŽđ”đ¶đ· with đŽđ¶ (see Figure 4).
Figure 4.
Since triangles PDC and PAD are alike, we
conclude that: đđ·
đđŽ=
đđ¶
đđ·=
đ·đ¶
đŽđ· . (5)
From relations (5), we find that:
đđŽ
đđ¶= (
đŽđ·
đ·đ¶)2. (6)
Ion Patrascu, Florentin Smarandache
144
This relationship indicates that P is the harmonic
conjugate of K with respect to A and C, so đ·đ is an
external simedian from D of the triangle đŽđ·đ¶.
Similarly, if we denote by đâ the intersection of
the tangent taken in B to the circle circumscribed with
đŽđ¶, we get:
đâČđŽ
đâČđ¶= (
đ”đŽ
đ”đ¶)2. (7)
From (6) and (7), as well as from the properties
of the harmonic quadrilateral, we know that: đŽđ”
đ”đ¶=
đŽđ·
đ·đ¶ ,
which means that:
đđŽ
đđ¶=
đâČđŽ
đâČđ¶ ,
hence đ = đâ.
Similarly, it is shown that the tangents taken to
A and C intersect at point Q located on the diagonal đ”đ·.
5th Remark.
a. The points P and Q are the diagonal poles of
đ”đ· and đŽđ¶ in relation to the circle
circumscribed to the quadrilateral.
b. From the previous Proposition, it follows that
in a triangle the internal simedian of an angle
is consecutive to the external simedians of
the other two angles.
Complements to Classic Topics of Circles Geometry
145
Figure 5.
8th Proposition.
Let đŽđ”đ¶đ· be an harmonic quadrilateral inscribed
in the circle of center O and let P and Q be the
intersections of the tangents taken in B and D,
respectively in A and C to the circle circumscribed to
the quadrilateral.
If {đŸ} = đŽđ¶ â© đ”đ·, then the orthocenter of
triangle đđŸđ is O.
Proof.
From the properties of tangents taken from a
point to a circle, we conclude that đđ â„ đ”đ· and đđ â„
đŽđ¶. These relations show that in the triangle đđŸđ, đđ
and đđ are heights, so đ is the orthocenter of this
triangle.
Ion Patrascu, Florentin Smarandache
146
4th Definition.
The Apolloniusâs circle related to the vertex A of
the triangle đŽđ”đ¶ is the circle built on the segment [đ·đž]
in diameter, where D and E are the feet of the internal,
respectively external, bisectors taken from A to the
triangle đŽđ”đ¶.
6th Remark.
If the triangle đŽđ”đ¶ is isosceles with đŽđ” = đŽđ¶ ,
the Apolloniusâs circle corresponding to vertex A is not
defined.
9th Proposition.
The Apolloniusâs circle relative to the vertex A of
the triangle đŽđ”đ¶ has as center the feet of the external
simedian taken from A.
Proof.
Let Oa be the intersection of the external
simedian of the triangle đŽđ”đ¶ with đ”đ¶ (see Figure 6).
Assuming that đ(ïżœïżœ) > đ(ïżœïżœ), we find that:
đ(đžđŽïżœïżœ) =1
2[đ(ïżœïżœ) + đ(ïżœïżœ)].
Oa being a tangent, we find that đ(đđđŽïżœïżœ) = đ(ïżœïżœ).
Withal,
đ(đžđŽđđ) =1
2[đ(ïżœïżœ) â đ(ïżœïżœ)]
Complements to Classic Topics of Circles Geometry
147
and
đ(đŽđžđđ) =1
2[đ(ïżœïżœ) â đ(ïżœïżœ)].
It results that đđđž = đđđŽ; onward, đžđŽđ· being a
right angled triangle, we obtain: đđđŽ = đđđ·; hence Oa
is the center of Apolloniusâs circle corresponding to
the vertex đŽ.
Figura 6.
10th Proposition.
Apolloniusâs circle relative to the vertex A of
triangle đŽđ”đ¶ cuts the circle circumscribed to the
triangle following the internal simedian taken from A.
Proof.
Let S be the second point of intersection of
Apolloniusâs Circle relative to vertex A and the circle
circumscribing the triangle đŽđ”đ¶.
Ion Patrascu, Florentin Smarandache
148
Because đđđŽ is tangent to the circle circum-
scribed in A, it results, for reasons of symmetry, that
đđđ will be tangent in S to the circumscribed circle.
For triangle đŽđ¶đ , đđđŽ and đđđ are external
simedians; it results that đ¶đđ is internal simedian in
the triangle đŽđ¶đ , furthermore, it results that the
quadrilateral đŽđ”đđ¶ is an harmonic quadrilateral.
Consequently, đŽđ is the internal simedian of the
triangle đŽđ”đ¶ and the property is proven.
7th Remark.
From this, in view of Figure 5, it results that the
circle of center Q passing through A and C is an
Apolloniusâs circle relative to the vertex A for the
triangle đŽđ”đ·. This circle (of center Q and radius QC)
is also an Apolloniusâs circle relative to the vertex C of
the triangle đ”đ¶đ·.
Similarly, the Apolloniusâs circle corresponding
to vertexes B and D and to the triangles ABC, and ADC
respectively, coincide.
We can now formulate the following:
11th Proposition.
In an harmonic quadrilateral, the Apolloniusâs
circle - associated with the vertexes of a diagonal and
to the triangles determined by those vertexes to the
other diagonal - coincide.
Complements to Classic Topics of Circles Geometry
149
Radical axis of the Apolloniusâs circle is the right
determined by the center of the circle circumscribed
to the harmonic quadrilateral and by the intersection
of its diagonals.
Proof.
Referring to Fig. 5, we observe that the power of
O towards the Apolloniusâs Circle relative to vertexes
B and C of triangles đŽđ”đ¶ and đ”đ¶đ is:
đđ”2 = đđ¶2.
So O belongs to the radical axis of the circles.
We also have đŸđŽ â đŸđ¶ = đŸđ” â đŸđ· , relatives
indicating that the point K has equal powers towards
the highlighted Apolloniusâs circle.
References.
[1] Roger A. Johnson: Advanced Euclidean Geometry,
Dover Publications, Inc. Mineola, New York,
USA, 2007.
[2] F. Smarandache, I. Patrascu: The Geometry of
Homological Triangles, The Education Publisher,
Inc. Columbus, Ohio, USA, 2012.
[2] F. Smarandache, I. Patrascu: Variance on Topics of
plane Geometry, The Education Publisher, Inc.
Columbus, Ohio, USA, 2013.
Ion Patrascu, Florentin Smarandache
150
Complements to Classic Topics of Circles Geometry
151
Triangulation of a Triangle
with Triangles having Equal
Inscribed Circles
In this article, we solve the following
problem:
Any triangle can be divided by a
cevian in two triangles that have
congruent inscribed circles.
Solution.
We consider a given triangle đŽđ”đ¶ and we show
that there is a point đ· on the side (đ”đ¶) so that the
inscribed circles in the triangles đŽđ”đ· , đŽđ¶đ· are
congruent. If đŽđ”đ¶ is an isosceles triangle (đŽđ” = đŽđ¶),
where đ· is the middle of the base (đ”đ¶), we assume
that đŽđ”đ¶ is a non-isosceles triangle. We note đŒ1, đŒ2 the
centers of the inscribed congruent circles; obviously,
đŒ1đŒ2 is parallel to the đ”đ¶. (1)
We observe that:
đ(âąđŒ1đŽđŒ2) =1
2đ(ïżœïżœ). (2)
Ion Patrascu, Florentin Smarandache
152
Figure 1.
If đ1, đ2 are contacts with the circles of the cevian
đŽđ· , we have â đŒ1đ1đ ⥠â đŒ2đ2đ ; let đ be the
intersection of đŒ1đŒ2 with đŽđ·, see Figure 1.
From this congruence, it is obvious that:
(đŒ1đ) ⥠(đŒ2đ). (3)
Let đŒ be the center of the circle inscribed in the
triangle đŽđ”đ¶; we prove that: đŽđŒ is a simedian in the
triangle đŒ1đŽđŒ2. (4)
Indeed, noting đŒ = đ(đ”đŽđŒ1) , it follows that
đ(âąđŒ1đŽđ) = đŒ. From âąđŒ1đŽđŒ2 = âąđ”đŽđŒ, it follows that
âąđ”đŽđŒ1 ⥠âąđŒđŽđŒ2 , therefore âąđŒ1đŽđ ⥠âąđŒđŽđŒ2 , indicating
that đŽđ and đŽđŒ are isogonal cevians in the triangle
đŒ1đŽđŒ2.
Since in this triangle đŽđ is a median, it follows
that đŽđŒ is a bimedian.
Complements to Classic Topics of Circles Geometry
153
Now, we show how we build the point đ·, using
the conditions (1) â (4), and then we prove that this
construction satisfies the enunciation requirements.
Building the point D.
10: We build the circumscribed circle of the given
triangle đŽđ”đ¶; we build the bisector of the
angle đ”đŽđ¶ and denote by P its intersection
with the circumscribed circle (see Figure
2).
20: We build the perpendicular on đ¶ to đ¶đ and
(đ”đ¶) side mediator; we denote đ1 the
intersection of these lines.
30: We build the circle â(đ1; đ1đ¶) and denote đŽâ
the intersection of this circle with the
bisector đŽđŒ (đŽâ is on the same side of the
line đ”đ¶ as đŽ).
40: We build through đŽ the parallel to đŽâđ1 and
we denote it đŒđ1.
50: We build the circle â(đ1âČ ; đ1
âČđŽ) and we denote
đŒ1, đŒ2 its intersections with đ”đŒ , and đ¶đŒ
respectively.
60: We build the middle đ of the segment (đŒ1đŒ2)
and denote by đ· the intersection of the
lines đŽđ and đ”đ¶.
Ion Patrascu, Florentin Smarandache
154
Figure 2.
Proof.
The point đ is the middle of the arc đ”ïżœïżœ , then
đ(đđ¶ïżœïżœ) = 1
2đ(ïżœïżœ).
The circle â(đ1; đ1đ¶) contains the arc from
which points the segment (BC) âis shownâ under angle
measurement 1
2đ(ïżœïżœ).
The circle â(đ1âČ ; đ1
âČđŽ) is homothetical to the
circle â(đ1; đ1đ¶) by the homothety of center đŒ and by
the report đŒđŽâČ
đŒđŽ; therefore, it follows that đŒ1đŒ2 will be
parallel to the đ”đ¶ , and from the points of circle
â(đ1âČ ; đ1
âČđŽ) of the same side of đ”đ¶ as đŽ, the segment
Complements to Classic Topics of Circles Geometry
155
(đŒ1đŒ2) âis shownâ at an angle of measure 1
2đ(ïżœïżœ). Since
the tangents taken in đ” and đ¶ to the circle â(đ1; đ1đ¶)
intersect in đ, on the bisector đŽđŒ, as from a property
of simedians, we get that đŽâČđŒ is a simedian in the
triangle đŽâČđ”đ¶.
Due to the homothetical properties, it follows
also that the tangents in the points đŒ1, đŒ2 to the circle
â(đ1âČ ; đ1
âČđŽ) intersect in a point đâ located on đŽđŒ, i.e. đŽđâ
contains the simedian (đŽđ) of the triangle đŒ1đŽđŒ2, noted
{đ} = đŽđâČ â© đŒ1đŒ2.
In the triangle đŒ1đŽđŒ2, đŽđ is a median, and đŽđ is
simedian, therefore âąđŒ1đŽđ ⥠đŒ2đŽđ; on the other hand,
âąđ”đŽđ ⥠âąđŒ1đŽđŒ2; it follows that âąđ”đŽđŒ1 ⥠đŒ2đŽđ, and more:
âąđŒ1đŽđ ⥠đ”đŽđŒ1, which shows that đŽđŒ1 is a bisector in the
triangle đ”đŽđ·; plus, đŒ1, being located on the bisector of
the angle đ”, it follows that this point is the center of
the circle inscribed in the triangle đ”đŽđ·.
Analogous considerations lead to the conclusion
that đŒ2 is the center of the circle inscribed in the
triangle đŽđ¶đ·. Because đŒ1đŒ2 is parallel to đ”đ¶, it follows
that the rays of the circles inscribed in the triangles
đŽđ”đ· and đŽđ¶đ· are equal.
Discussion.
The circles â(đ1; đ1đŽâČ), â(đ1
âČ ; đ1âČđŽ) are unique;
also, the triangle đŒ1đŽđŒ2 is unique; therefore,
determined as before, the point đ· is unique.
Ion Patrascu, Florentin Smarandache
156
Remark.
At the beginning of the Proof, we assumed that
đŽđ”đ¶ is a non-isosceles triangle with the stated
property. There exist such triangles; we can construct
such a triangle starting "backwards". We consider two
given congruent external circles and, by tangent
constructions, we highlight the đŽđ”đ¶ triangle.
Open problem.
Given a scalene triangle đŽđ”đ¶ , could it be
triangulated by the cevians đŽđ· , đŽđž , with đ· , đž
belonging to (đ”đ¶), so that the inscribed circles in the
triangles đŽđ”đ·, đ·đŽđž and the đžđŽđ¶ to be congruent?
Complements to Classic Topics of Circles Geometry
157
An Application of a Theorem
of Orthology
In this short article, we make connections
between Problem 21 of [1] and the theory
of orthological triangles.
The enunciation of the competitional problem is:
Let (đđŽ), (đđ”), (đđ¶) be the tangents in the
peaks đŽ, đ”, đ¶ of the triangle đŽđ”đ¶ to the circle
circumscribed to the triangle. Prove that the
perpendiculars drawn from the middles of the
opposite sides on (đđŽ), (đđ”), (đđ¶) are concurrent
and determine their concurrent point.
We formulate and we demonstrate below a
sentence containing in its proof the solution of the
competitional problem in this case.
Proposition.
The tangential triangle and the median triangle
of a given triangle đŽđ”đ¶ are orthological. The
orthological centers are đ â the center of the circle
circumscribed to the triangle đŽđ”đ¶, and đ9 â the center
of the circle of đŽđ”đ¶ triangleâs 9 points.
Ion Patrascu, Florentin Smarandache
158
Figure 1.
Proof.
Let đđđđđđ be the median triangle of triangle
đŽđ”đ¶ and đđđđđđ the tangential triangle of the triangle
đŽđ”đ¶ . It is obvious that the triangle đđđđđđ and the
triangle đŽđ”đ¶ are orthological and that đ is the
orthological center.
Verily, the perpendiculars taken from đđ, đđ , đđ
on đ”đ¶ ; đ¶đŽ; đŽđ” respectively are internal bisectors in
the triangle đđđđđđ and consequently passing through
đ, which is center of the circle inscribed in triangle
đđđđđđ . Moreover, đđđ is the mediator of (đ”đ¶) and
accordingly passing through đđ , and đđđđ is
perpendicular on đ”đ¶ , being a mediator, but also on
đđđđ which is a median line.
Complements to Classic Topics of Circles Geometry
159
From orthological triangles theorem, it follows
that the perpendiculars taken from đđ , đđ , đđ on đđđđ,
đđđđ , đđđđ respectively, are concurrent. The point of
concurrency is the second orthological center of
triangles đđđđđđ and đđđđđđ. We prove that this point
is đ9 â the center of Euler circle of triangle đŽđ”đ¶. We
take đđđ1 â„ đđđđ and denote by {đ»1} = đđđ1 â© đŽđ», đ»
being the orthocenter of the triangle đŽđ”đ¶. We know
that đŽđ» = 2đđđ; we prove this relation vectorially.
From Sylvesterâs relation, we have that: đđ» =
đđŽ + đđ” + đđ¶ , but đđ” + đđ¶ = 2đđđ ; it follows that
đđ» â đđŽ = 2đđđ , so đŽđ» = 2đđđ
; changing to module,
we have đŽđ» = 2đđđ . Uniting đ to đŽ , we have đđŽ â„
đđđđ , and because đđđ1 â„ đđđđ and đŽđ» â„ đđđ , it
follows that the quadrilateral đđđđ»1đŽ is a
parallelogram.
From đŽđ»1 = đđđ and đŽđ» = 2đđđ we get that đ»1
is the middle of (đŽđ»), so đ»1 is situated on the circle of
the 9 points of triangle đŽđ”đ¶. On this circle, we find as
well the points đŽâČ - the height foot from đŽ and đđ ;
since âąđŽđŽâČđđ = 900 , it follows that đđđ»1 is the
diameter of Euler circle, therefore the middle of
(đđđ»1), which we denote by đ9, is the center of Eulerâs
circle; we observe that the quadrilateral đ»1đ»đđđ is as
well a parallelogram; it follows that đ9 is the middle
of segment [đđ»].
In conclusion, the perpendicular from đđ on đđđđ
pass through đ9.
Ion Patrascu, Florentin Smarandache
160
Analogously, we show that the perpendicular
taken from đđ on đđđđ pass through đ9 and
consequently đ9 is the enunciated orthological center.
Remark.
The triangles đđđđđđ and đđđđđđ are
homological as well, since đđđđ , đđđđ , đđđđ are
concurrent in O, as we already observed, therefore the
triangles đđđđđđ and đđđđđđ are orthohomological of
rank I (see [2]).
From P. Sondat theorem (see [4]), it follows that
the Euler line đđ9 is perpendicular on the homological
axis of the median triangle and of the tangential
triangle corresponding to the given triangle đŽđ”đ¶.
Note.
(Regarding the triangles that are simultaneously
orthological and homological)
In the article A Theorem about Simultaneous
Orthological and Homological Triangles, by Ion
Patrascu and Florentin Smarandache, we stated and
proved a theorem which was called by Mihai Dicu in
[2] and [3] The Smarandache-Patrascu Theorem of
Orthohomological Triangle; then, in [4], we used the
term ortohomological triangles for the triangles that
are simultaneously orthological and homological.
The term ortohomological triangles was used by
J. Neuberg in Nouvelles Annales de Mathematiques
Complements to Classic Topics of Circles Geometry
161
(1885) to name the triangles that are simultaneously
orthogonal (one has the sides perpendicular to the
sides of the other) and homological.
We suggest that the triangles that are
simultaneously orthogonal and homological to be
called ortohomological triangles of first rank, and
triangles that are simultaneously orthological and
homological to be called ortohomological triangles of
second rank.
Ion Patrascu, Florentin Smarandache
162
References.
[1] Titu Andreescu and Dorin Andrica: 360 de probleme de matematicÄ pentru concursuri [360 Competitional Math Problem], Editura GIL,
Zalau, Romania, 2002. [2] Mihai Dicu, The Smarandache-Patrascu Theorem of
Orthohomological Triangle,
http://www.scrib.com/ doc/28311880/, 2010 [3] Mihai Dicu, The Smarandache-Patrascu Theorem of
Orthohomological Triangle, in âRevista de matematicÄ ALPHAâ, Year XVI, no. 1/2010, Craiova, Romania.
[4] Ion Patrascu, Florentin Smarandache: Variance on topics of Plane Geometry, Educational Publishing, Columbus, Ohio, 2013.
[5] Multispace and multistructure neutrosophic transdisciplinarity (100 collected papers of
sciences), Vol IV, Edited by prof. Florentin Smarandache, Dept. of Mathematics and Sciences, University of New Mexico, USA -
North European Scientific Publishers, Hanco, Finland, 2010.
Complements to Classic Topics of Circles Geometry
163
The Dual of a Theorem
Relative to the Orthocenter
of a Triangle
In [1] we introduced the notion of
Bobillierâs transversal relative to a point
đ¶ in the plane of a triangle đšđ©đȘ; we use
this notion in what follows.
We transform by duality with respect to a
circle â(đ, đ) the following theorem
relative to the orthocenter of a triangle.
1st Theorem.
If đŽđ”đ¶ is a nonisosceles triangle, đ» its
orthocenter, and AA1, BB1, CC1 are cevians of a triangle
concurrent at point đ different from đ», and đ,đ, đ are
the intersections of the perpendiculars taken from đ»
on given cevians respectively, with đ”đ¶, đ¶đŽ, đŽđ”, then
the points đ,đ, đ are collinear.
Ion Patrascu, Florentin Smarandache
164
Proof.
We note with đŒ = đ(âąBAA1); đœ = đ(âąCBB1), đŸ =
đ(âąACC1), see Figure 1.
According to Cevaâs theorem, trigonometric form,
we have the relation: sinđŒ
sin(đŽâđŒ)â
sinđœ
sin(đ”âđœ)â
sinđŸ
sin(đ¶âđŸ)= 1. (1)
We notice that: đđ”
đđ¶=
Area(đđ»đ”)
Area(đđ»đ¶)=
đđ»âđ»đ”âsin sin(đđ»ïżœïżœ)
đđ»âđ»đ¶âsin(đđ»ïżœïżœ) .
Because:
âąđđ»đ” ⥠âąđŽ1đŽđ¶,
as angles of perpendicular sides, it follows that
đ(âąđđ»đ”) = đ(ïżœïżœ) â đŒ.
Therewith:
đ(âąđđ»đ¶) = đ(đđ»ïżœïżœ) + đ(đ”đ»ïżœïżœ) = 1800đŒ.
We thus get that:
đđ”
đđ¶=
sin(đŽ â đŒ)
sinđŒâđ»đ”
đ»đ¶.
Analogously, we find that: đđ¶
đđŽ=
sin(đ”âđœ)
sinđœâđ»đ¶
đ»đŽ ;
đđŽ
đđ”=
sin(đ¶âđŸ)
sinđŸâđ»đŽ
đ»đ” .
Applying the reciprocal of Menelaus' theorem,
we find, in view of (1), that: đđ”
đđ¶âđ»đ¶
đ»đŽâđđŽ
đđ”= 1.
This shows that đ,đ, đ are collinear.
Complements to Classic Topics of Circles Geometry
165
Figure 1.
Note.
1st Theorem is true even if đŽđ”đ¶ is an obtuse,
nonisosceles triangle.
The proof is adapted analogously.
2nd Theorem. (The Dual of the Theorem 1)
If đŽđ”đ¶ is a triangle, đ a certain point in his plan,
and đŽ1, đ”1, đ¶1 Bobillierâs transversals relative to đ of
Ion Patrascu, Florentin Smarandache
166
đŽđ”đ¶ triangle, as well as đŽ2 â đ”2 â đ¶2 a certain
transversal in đŽđ”đ¶, and the perpendiculars in đ, and
on đđŽ2, đđ”2, đđ¶2 respectively, intersect the Bobillierâs
transversals in the points đŽ3, đ”3, đ¶3, then the ceviens
đŽđŽ3, đ”đ”3, đ¶đ¶3 are concurrent.
Proof.
We convert by duality with respect to a circle
â(đ, đ) the figure indicated by the statement of this
theorem, i.e. Figure 2. Let đ, đ, đ be the polars of the
points đŽ, đ”, đ¶ with respect to the circle â(đ, đ). To the
lines đ”đ¶, đ¶đŽ, đŽđ” will correspond their poles {đŽâČ4 =
đđđ; {đ”âČ4 = đđđ; {đ¶âČ4 = đđđ.
To the points đŽ1, đ”1, đ¶1 will respectively
correspond their polars đ1, đ1, đ1 concurrent in
transversalâs pole đŽ1 â đ”1 â đ¶1.
Since đđŽ1 â„ đđŽ, it means that the polars đ and đ1
are perpendicular, so đ1 â„ đ”âČđ¶âČ, but đ1 pass through đŽâČ,
which means that đâČ contains the height from đŽâČ of
đŽâČđ”âČđ¶âČ triangle and similarly đ1 contains the height
from đ”âČ and đ1 contains the height from đ¶âČ of đŽâČđ”âČđ¶âČ
triangle.
Consequently, the pole of đŽ1 â đ”1 â đ¶1
transversal is the orthocenter đ»âČ of đŽâČđ”âČđ¶âČ triangle. In
the same way, to the points đŽ2, đ”2, đ¶2 will correspond
the polars to đ2, đ2, đ2 which pass respectively through
đŽâČ, đ”âČ, đ¶âČ and are concurrent in a point đâČ, the pole of
the line đŽ2 â đ”2 â đ¶2 with respect to the circle â(đ, đ).
Complements to Classic Topics of Circles Geometry
167
Figure 2.
Given đđŽ2 â„ đđŽ3, it means that the polar đ2 and
đ3 are perpendicular, đ2 correspond to the cevian
đŽâČđâČ, also đ3 passes through the the pole of the
transversal đŽ1 â đ”1 â đ¶1, so through đ»âČ, in other words
đ3 is perpendicular taken from đ»âČ on đŽâČđâČ; similarly,
đ2 â„ đ3, đ2 â„ đ3, so đ3 is perpendicular taken from đ»âČ
on đ¶âČđâČ. To the cevian đŽđŽ3 will correspond by duality
considered to its pole, which is the intersection of the
polars of đŽ and đŽ3, i.e. the intersection of lines đ and
đ3 , namely the intersection of đ”âČđ¶âČ with the
perpendicular taken from đ»âČ on đŽâČđâČ; we denote this
Ion Patrascu, Florentin Smarandache
168
point by đâČ. Analogously, we get the points đâČ and đâČ.
Thereby, we got the configuration from 1st Theorem
and Figure 1, written for triangle đŽâČđ”âČđ¶âČ of orthocenter
đ»âČ. Since from 1st Theorem we know that đâČ, đâČ, đâČ are
collinear, we get the the cevians đŽđŽ3, đ”đ”3, đ¶đ¶3 are
concurrent in the pole of transversal đâČ â đâČ â đâČ with
respect to the circle â(đ, đ), and 2nd Theorem is proved.
References.
[1] Ion Patrascu, Florentin Smarandache: The Dual Theorem concerning Aubertâs Line, below.
[2] Florentin Smarandache, Ion Patrascu: The
Geometry of Homological Triangles. The Education Publisher Inc., Columbus, Ohio, USA
â 2012.
Complements to Classic Topics of Circles Geometry
169
The Dual Theorem Concerning
Aubertâs Line
In this article we introduce the concept of
Bobillierâs transversal of a triangle with
respect to a point in its plan ; we prove the
Aubertâs Theorem about the collinearity of
the orthocenters in the triangles
determined by the sides and the diagonals
of a complete quadrilateral, and we obtain
the Dual Theorem of this Theorem.
1st Theorem. (E. Bobillier)
Let đŽđ”đ¶ be a triangle and đ a point in the plane
of the triangle so that the perpendiculars taken in đ,
and đđŽ,đđ”,đđ¶ respectively, intersect the sides
đ”đ¶, đ¶đŽ and đŽđ” at đŽđ, đ”đ Ći đ¶đ. Then the points đŽđ,
đ”đ and đ¶đ are collinear.
Proof.
We note that đŽđđ”
đŽđđ¶=
aria (đ”đđŽđ)
aria (đ¶đđŽđ) (see Figure 1).
Ion Patrascu, Florentin Smarandache
170
Figure 1.
Area (đ”đđŽđ) =1
2â đ”đ â đđŽđ â sin(đ”đđŽïżœïżœ).
Area (đ¶đđŽđ) =1
2â đ¶đ â đđŽđ â sin(đ¶đđŽïżœïżœ).
Since
đ(đ¶đđŽïżœïżœ) =3đ
2â đ(đŽđïżœïżœ),
it explains that
sin(đ¶đđŽïżœïżœ) = â cos(đŽđïżœïżœ);
sin(đ”đđŽïżœïżœ) = sin (đŽđïżœïżœ âđ
2) = â cos(đŽđïżœïżœ).
Therefore:
đŽđđ”
đŽđđ¶=
đđ” â cos(đŽđïżœïżœ)
đđ¶ â cos(đŽđïżœïżœ) (1).
In the same way, we find that:
Complements to Classic Topics of Circles Geometry
171
đ”đđ¶
đ”đđŽ=
đđ¶
đđŽâcos(đ”đïżœïżœ)
cos(đŽđïżœïżœ) (2);
đ¶đđŽ
đ¶đđ”=
đđŽ
đđ”âcos(đŽđïżœïżœ)
cos(đ”đïżœïżœ) (3).
The relations (1), (2), (3), and the reciprocal
Theorem of Menelaus lead to the collinearity of points
đŽđ,đ”đ, đ¶đ.
Note.
Bobillier's Theorem can be obtained â by
converting the duality with respect to a circle â from
the theorem relative to the concurrency of the heights
of a triangle.
1st Definition.
It is called Bobillierâs transversal of a triangle
đŽđ”đ¶ with respect to the point đ the line containing
the intersections of the perpendiculars taken in đ on
đŽđ, đ”đ, and đ¶đ respectively, with sides đ”đ¶, CA and
đŽđ”.
Note.
The Bobillierâs transversal is not defined for any
point đ in the plane of the triangle đŽđ”đ¶, for example,
where đ is one of the vertices or the orthocenter đ» of
the triangle.
Ion Patrascu, Florentin Smarandache
172
2nd Definition.
If đŽđ”đ¶đ· is a convex quadrilateral and đž, đč are the
intersections of the lines đŽđ” and đ¶đ· , đ”đ¶ and đŽđ·
respectively, we say that the figure đŽđ”đ¶đ·đžđč is a
complete quadrilateral. The complete quadrilateral
sides are đŽđ”, đ”đ¶, đ¶đ·, đ·đŽ , and đŽđ¶, đ”đ· and đžđč are
diagonals.
2nd Theorem. (Newton-Gauss)
The diagonalsâ middles of a complete
quadrilateral are three collinear points. To prove 2nd
theorem, refer to [1].
Note.
It is called Newton-Gauss Line of a quadrilateral
the line to which the diagonalsâ middles of a complete
quadrilateral belong.
3rd Theorem. (Aubert)
If đŽđ”đ¶đ·đžđč is a complete quadrilateral, then the
orthocenters đ»1, đ»2, đ»3, đ»4 of the traingles đŽđ”đč, đŽđžđ·,
đ”đ¶đž, and đ¶đ·đč respectively, are collinear points.
Complements to Classic Topics of Circles Geometry
173
Proof.
Let đŽ1, đ”1, đč1 be the feet of the heights of the
triangle đŽđ”đč and đ»1 its orthocenter (see Fig. 2).
Considering the power of the point đ»1 relative to the
circle circumscribed to the triangle ABF, and given the
triangle orthocenterâs property according to which its
symmetrics to the triangle sides belong to the
circumscribed circle, we find that:
đ»1đŽ â đ»1đŽ1 = đ»1đ” â đ»1đ”1 = đ»1đč â đ»1đč1.
Figure 2.
This relationship shows that the orthocenter đ»1
has equal power with respect to the circles of
diameters [đŽđ¶], [đ”đ·], [đžđč]. As well, we establish that
the orthocenters đ»2, đ»3, đ»4 have equal powers to these
circles. Since the circles of diameters [đŽđ¶], [đ”đ·], [đžđč]
have collinear centers (belonging to the Newton-
Ion Patrascu, Florentin Smarandache
174
Gauss line of the đŽđ”đ¶đ·đžđč quadrilateral), it follows
that the points đ»1, đ»2, đ»3, đ»4 belong to the radical axis
of the circles, and they are, therefore, collinear points.
Notes.
1. It is called the Aubert Line or the line of the
complete quadrilateralâs orthocenters the line to
which the orthocenters đ»1, đ»2, đ»3, đ»4 belong.
2. The Aubert Line is perpendicular on the
Newton-Gauss line of the quadrilateral (radical axis of
two circles is perpendicular to their centersâ line).
4th Theorem. (The Dual Theorem of the 3rd Theorem)
If đŽđ”đ¶đ· is a convex quadrilateral and đ is a
point in its plane for which there are the Bobillierâs
transversals of triangles đŽđ”đ¶ , đ”đ¶đ· , đ¶đ·đŽ and đ·đŽđ” ;
thereupon these transversals are concurrent.
Proof.
Let us transform the configuration in Fig. 2, by
duality with respect to a circle of center đ.
By the considered duality, the lines đ, đ, đ, đ, đ Ći
đ correspond to the points đŽ, đ”, đ¶, đ·, đž, đč (their polars).
It is known that polars of collinear points are
concurrent lines, therefore we have: đ â© đ â© đ = {đŽâČ},
Complements to Classic Topics of Circles Geometry
175
đ â© đ â© đ = {đ”âČ} , đ â© đ â© đ = {đ¶âČ} , đ â© đ â© đ = {đ·âČ} , đ â©
đ = {đžâČ}, đ â© đ = {đčâČ}.
Consequently, by applicable duality, the points
đŽâČ, đ”âČ, đ¶âČ, đ·âČ, đžâČ and đčâČ correspond to the straight lines
đŽđ”, đ”đ¶, đ¶đ·, đ·đŽ, đŽđ¶, đ”đ·.
To the orthocenter đ»1 of the triangle đŽđžđ· , it
corresponds, by duality, its polar, which we denote
đŽ1âČ â đ”1
âČ â đ¶1âČ, and which is the Bobillierâs transversal of
the triangle đŽâđ¶âđ·â in relation to the point đ. Indeed,
the point đ¶â corresponds to the line đžđ· by duality; to
the height from đŽ of the triangle đŽđžđ·, also by duality,
it corresponds its pole, which is the point đ¶1âČ located
on đŽâđ·â such that đ(đ¶âČđđ¶1âČ ) = 900.
To the height from đž of the triangle đŽđžđ· , it
corresponds the point đ”1âČ â đŽâČđ¶âČ such that đ(đ·âČđđ”1
âČ ) =
900.
Also, to the height from đ·, it corresponds đŽ1âČ â
đ¶âČđ·â on C such that đ(đŽâČđđŽ1âČ ) = 900. To the
orthocenter đ»2 of the triangle đŽđ”đč, it will correspond,
by applicable duality, the Bobillierâs transversal đŽ2âČ â
đ”2âČ â đ¶2
âČ in the triangle đŽâČđ”âČđ·âČ relative to the point đ.
To the orthocenter đ»3 of the triangle đ”đ¶đž , it will
correspond the Bobillierâs transversal đŽ3âČ â đ”âČ3 â đ¶3âČ
in the triangle đŽâČđ”âČđ¶âČ relative to the point đ, and to
the orthocenter đ»4 of the triangle đ¶đ·đč , it will
correspond the transversal đŽ4âČ â đ”4
âČ â đ¶4âČ in the triangle
đ¶âČđ·âČđ”âČ relative to the point đ.
Ion Patrascu, Florentin Smarandache
176
The Bobillierâs transversals đŽđâČ â đ”đ
âČ â đ¶đâČ , đ = 1,4
correspond to the collinear points đ»đ, đ = 1,4 .
These transversals are concurrent in the pole of
the line of the orthocenters towards the considered
duality.
It results that, given the quadrilateral đŽâđ”âđ¶âđ·â,
the Bobillierâs transversals of the triangles đŽâđ¶âđ·â ,
đŽâđ”âđ·â , đŽâđ”âđ¶â and đ¶âđ·âđ”â relative to the point đ are
concurrent.
References.
[1] Florentin Smarandache, Ion Patrascu: The Geometry of Homological Triangles. The
Education Publisher Inc., Columbus, Ohio, USA â 2012.
[2] Ion Patrascu, Florentin Smarandache: Variance on Topics of Plane Geometry. The Education Publisher Inc., Columbus, Ohio, USA â 2013.
Complements to Classic Topics of Circles Geometry
177
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Ion Patrascu, Florentin Smarandache
180
We approach several themes of clas-
sical geometry of the circle and complete
them with some original results, showing
that not everything in traditional math is
revealed, and that it still has an open
character.
The topics were chosen according to
authorsâ aspiration and attraction, as a
poet writes lyrics about spring according to
his emotions.