triangle centers frank koegel summer institute 2007
TRANSCRIPT
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Triangle Centers
Frank KoegelSummer Institute 2007
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What are the properties of a median in a triangle?
A median in a triangle is the segment that joins a vertex with the midpoint of the opposite side.
There are three medians in a triangle.
How many medians are in a triangle?
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Let’s use GeoGebra to create medians in triangles!
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Medians are concurrent
The medians in a triangle are concurrent (i.e., they meet in one interior point of the triangle.) centroid
Mc
Mb
Ma
A
C
B
The point of concurrency is the centroid of a triangle.
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Are the six triangles formed by the medians
similar? The medians
split the triangle in six smaller
triangles. Mc
centroid
Ma
Mb
A
C
B
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Mc
centroid
A
C
B
Do the blue and the yellow triangles have the same area?
A=xA=x
The distance from the centroid to AB is the –common--height for both triangles.
(remember that Mc is a midpoint)
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Mc
A
B
C
Notice that triangles CAMc and CMcB have the same area as well (again the base is the same and the height is the same).
A=yA=y
A=zA=z
A=xA=x
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Mc
A
B
C
So the orange triangle and the pink triangle have the same area (since we already proved that the yellow triangle and the blue triangle have the same area).
A=xA=x
A=z/2
A=z/2
A=z/2
A=z/2
And this in turn shows that all six little triangles have the same area.
A=zA=z
We can divide the orange and pink into two congruent halves for the same reason we were able to divide triangle AGB
G
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Let’s use GeoGebra to create angle bisectors with triangles!
Angle Bisectors
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Angle bisectors are concurrent
• The angle bisectors in a triangle are concurrent (i.e., they meet in one interior point of the triangle.)
• The point of concurrency is the incenter of a triangle.
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The Angle Bisectors in a Triangle
The incenter is the center of a triangle's incircle. It can be found as the intersection of the angle bisectors.
(A angle bisector is a line that bisects an angle into two congruent triangles.)
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GeoGebra
Let’s look at a file on
using a circle on a segment.
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The Perpendicular Bisectors in a Triangle
The circumcenter is the center of a triangle's circumcircle. It can be found as the intersection of the perpendicular bisectors.
(A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint.)
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Perpendicular bisectors are concurrent
• The perpendicular bisectors in a triangle are concurrent (i.e., they meet in one interior point of the triangle.)
• The point of concurrency is the circumcenter of a triangle.
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The circumcenter is equidistant from any pair of the triangle's points, and all points on the perpendicular bisectors are equidistant from those points of the triangle.
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In the special case of a right triangle, the circumcenter (C in the figure at right) lies exactly at the midpoint of the hypotenuse (longest side).
Where would the circumcenter beon a right triangle?
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A triangle is acute (all angles smaller than a right angle) if and only if the circumcenter lies inside the triangle; it is obtuse (has an angle bigger than a right one) if and only if the circumcenter lies outside, and it is a right triangle if and only if the circumcenter lies on one of its sides (namely on the hypotenuse). This is one form of Thales' theorem.
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Let’s use GeoGebra to create altitudes with triangles!
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The Altitudes in a Triangle
The orthocenter is the intersection of the altitudes in a triangle.
(An altitude in a triangle is the perpendicular distance from a vertex to the base opposite.)
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Altitudes are concurrent
• The altitudes in a triangle are concurrent.
• The point of concurrency is the orthocenter of a triangle.
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Let’s use GeoGebra to create the Fermat Point!
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Fermat Point
The sum PA+PB+PC is thesmallest distance possiblefrom the three original vertices. (Angles must be less than 120 degrees)
The interior angles APB, BPC and APC are congruent
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• Let’s combine some of our constructions
• Our goal is to find all nine points on the 9 point circle
9 Point Circle
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We need 3 sets of points:We need 3 midpoints of the triangle.
We need the midpoints of the triangles vertices and the orthocenter.
We need 3 altitudes of the triangle.
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When you put any two of these points on a circle, you should find that all 9 are on that circle!
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Symmedian PointLet’s try to create this!
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What are the requirements for a triangle center?Homogeneity
Cyclicity (trilinear coordinates)
Bisymmetry
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Homogeneity
This refers to similar triangles having similarly placed centers.
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Cyclicity Trilinear coordinates
The reference point (P) has proportional directed distance from the center to the sideline.
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There are 27 widely accepted triangle centers, with 15 classical ones, and 12 more recent additions.
But according to Wolfram’s website, the most recent update shows 2676 centers.
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Sunshine State Standards
MA.C.2.4: The student visualizes and illustrates ways in which shapes can be combined, subdivided, and changed.
MA.C.2.4.1: The student understands geometric concepts such as perpendicularity, parallelism, tangency, congruency, similarity, reflections, symmetry, and transformations including flips, slides, turns, enlargements, rotations, and fractals.