cmbuk 1 segçbrubmnþsmxan;;² 1> tmnak;tmngkñúgrtiekan · 2 a cos 2 c cos 2 b acos 2 a a r...
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CMBUk 1 segçbrUbmnþsMxan;;²
1> TMnak;TMngkñúgRtIekaN
nimitþsBaØaEdleRbI³ a, b, c : rgVas;RCugEdlQmmMu A, B, C én ΔABC
S : RklaépÞRtIekaN p : knøHbrimaRténRtIekaN ha, hb, hc : RbEvgkMBs;KUsecjBIkMBUl A, B, C
la, lb, lc : RbEvgknøHbnÞat;BuHmMukñúg ma, mb, mc : RbEvgemdüanKUsecjBIkMBUlTaMgbI ra, rb, rc : RbEvgkaMrgVg; R : RbEvgkaMrgVg;carweRkARtIekaN ABC
r : kaMrgVg;carwkkñúg ΔABC
1>RTwsþIbTsuInIs³
R2Csin
cBsin
bAsin
a===
2> RTwsþIbTkUsuInIs³
1
Ccosab2bacBcosac2cabAcosbc2cba
222
222
222
−+=
−+=
−+=
3> RTwsþIbTcMenal³
AcosbBcosacCcosaAcoscbBcoscCcosba
+=+=+=
4. rUbmnþRklaépÞ³
Csin.Bsin.AsinR2
r)cp(r)bp(r)ap(prR4
abc
Csinab21Bsinac
21Asinbc
21
)cp)(bp)(ap(p
ch21bh
21ah
21S
2
cba
cba
=
−=−=−===
===
−−−=
===
5> rUbmnþkaMcarwkkñúg³
2Ctan)cp(
2Btan)bp(
2Atan)ap(
2Ccos
2Bsin
2Asinc
2Bcos
2Csin
2Asinb
2Acos
2Csin
2Bsina
r
−=−=−=
===
2
2Acos
2Ccos
2Bcosa
2Atanpra ==
2Bcos
2Ccos
2Acosb
2Btanprb ==
2Ccos
2Acos
2Bcosc
2Ctanprc ==
6> rUbmnþknøHbnÞat;BuHmMukñúg³
)ap(pbcca
2cb
2Asinbc2
la −+
=+
=
)bp(pacca
2ca
2Bcosac2
lb −+
=+
=
)cp(pabba
2ba
2Ccosab2
lc −+
=+
=
7> rUbmnþemdüan³
Ccosab2bacb2a2m4
Bcosca2acba2c2m4
Acosbc2cbac2b2m4
222222c
222222b
22222a
++=−+=
++=−+=
++=−+= 2
3
2> vismPaB
1> vismPaB Cauchy:
1. eK)an +∈∀ Rb,a ab2ba ≥+ . smPaBekIteLIgkalNa ba = .
2. CaTUeTA )n.......,3,2,1i(Rai =∈∀ eK)an ³ n321n321 a....aaana....aaa ≥++++
smPaBekIteLIgkalNa n321 a.......aaa ==== . 2> vismPaB Bernoulli:
1. eK)an³ ∗Ν∈−>∀ n,1x ( ) nx1x1 n +≥+ . smPaBekIteLIgkalNa 0x = rW 1n = . 2. 1r,Qr,1x ≥∈−>∀ eK)an ( ) rx1x1 r +≥+ . smPaBekIteLIgkalNa 0x = rW 1r = . 3. 1r1,Qr,1x ≤≤∈−>∀ eK)an ( ) rx1x1 r +≤+ . smPaBekIteLIgkalNa 0x = rW 1r = .
3> vismPaB Bunyakovsky:
1. eK)an³ Rd,c,b,a ∈∀ )dc)(ba(bdac 2222 ++≤+ rW )dc)(ba()bdac( 22222 ++≤+
smPaBekIteLIgkalNa dc
ba= .
4
2. )n,....,3,2,1i(Rb,a ii =∈∀ eK)an³ )b....bb)(a....aa()ba....baba( 2
n22
21
2n
22
21
2nn2211 ++++++≤+++
smPaBekIteLIgkalNa n
n
2
2
1
1
ba....
ba
ba
=== .
4> vismPaB Minkowski:
1. eK)an³ Rd,c,b,a ∈∀ 222222 dcba)db()ca( +++≤+++ .
smPaBekIteLIgkalNa dc
ba= .
2. )n,....,3,2,1i(Rb,a ii =∈∀ eK)an³
222
222
22
21
221
221
...
)...()...(
nn
nn
bababa
bbbaaa
++++++≤
+++++++
smPaBekIteLIgkalNa n
n
2
2
1
1
ba....
ba
ba
=== .
5> vismPaBRtIekaN³
ebI a, b, c CargVas;RCugénRtIekaN eK)anvismPaB³
1. cbacb +<<− 2. cabca +<<−
3. bacba +<<−
5
3> GnuKmn_RtIekaNmaRt
1> TMnak;TMngsMxan;²³
1. 1cossin 22 =α+α
2. α
=+α 22
cos11tan )0(cos ≠α
3. α
=+α 22
sin11cot )0(cos ≠α
2> rUbmnþplbUk nig pldk³ 1. bsinasinbcosacos)bacos( +=−
2. bsinasinbcosacos)bacos( −=+
3. acosbsinbcosasin)basin( +=+
4. acosbsinbcosasin)basin( −=−
5. btanatan1btanatan)batan(
−+
=+ ⎟⎠⎞
⎜⎝⎛ π+
π≠+ k
2)ba(,b,a
6. btanatan1btanatan)batan(
+−
=− ⎟⎠⎞
⎜⎝⎛ π+
π≠− k
2)ba(,b,a
3> rUbmnþmMu 2α nig 3α :
1. αα=α cossin22sin 2. 1cos22cos 2 −α=α
6
3. 4. α−=α 2sin212cos α−=α 22 sincos2cos
5. α−α
=α 2tan1tan22tan 6.
α−α
=αcot2
1cot2cot2
7. 8. α−α=α cos3cos43cos 3 α−α=α 3sin4sin33sin
9. α−α−α
=α 2
3
tan31tantan33tan 10.
1cot3cot3cot3cot 2
3
−αα−α
=α
4> rUbmnþknøHmMu³
1. 2cos1
2sin α−
±=α 2.
2cos1
2cos α+
±=α
3. α+α−
±=α
cos1cos1
2tan 4.
α+α
=α
cos1sin
2tan
5> KNna ααα tan,cos,sin CaGnuKmn_ )2
tant(t α=
22
2
2 t1t2tan,
t1t1cos,
t1t2sin
−=α
+−
=α+
=α
6> rUbmnþbMElg³
6>1 bMElgBIplKuNeTAplbUk
1. [ ])bacos()bacos(21bcosacos −++=
2. [ ])bacos()bacos(21bsinasin +−−=
7
3. [ ])basin()basin(21bcosasin −++=
4. [ ])basin()basin(21asinbsin −−+=
6>2 bMElgBIplbUkeTAplKuN
1. 2
qpcos2
qpcos2qcospcos −+=+
2. 2
qpsin2
qpsin2qcospcos −+−=−
3. 2
qpcos2
qpsin2qsinpsin −+=+
4. 2
qpcos2
qpsin2qsinpsin +−=−
5. qcospcos)qpsin(qtanptan +
=+
6. qcospcos)qpsin(qtanptan −
=−
7. qsinpsin)qpsin(qcotpcot +
=+
8. qsinpsin)pqsin(qcotpcot −
=−
8
4>bNþaRTwsþIbTepSgeTot
1> RTwsþIbTGgát;FñÚRbsBVKña³
ebIGgát;FñÚ AB & CD énrgVg;RbsBVKñaRtg; X enaH
XD.CXXB.AX =
2> RTwsþIbTknøHbnÞat;BuHmMu
ebI AX CaknøHbnÞat;BuH ∠A én ΔABC eK)an³
ACAB
XCBX
=
3> RTwsþIbT Ptolémé
ebI ABCD CactuekaNcarwkkñúgrgVg; nig man AC &
CD CaGgát;RTUg eK)an³
BD.ACAD.BCAB.CD =+
9
4> RTwsþIbT Céva:
ebI X, Y, Z CacMnucenAelIRCug BC, AC , AB én ΔABC Edl AX, BY, &
CZ RbsBVKñaRtg;cMnucEtmYy eK)an³
1YACY.
XCBX.
ZBAZ
=
(bMnkRsayenAlMhat;elx 66 )
5> RTwsþIbT Simson:
eKeGay P CacMnucenAelIrgVg;carwkeRkA ΔABC nig E
F, G CaeCIgcMenalEkgBI P eTAelIRCugTaMg3én ΔABC
eK)an³ E, F, G rt;Rtg;Kña.
(bMnkRsayenAlMhat;elx 67 )
10
CMBUk 2 lMhat; 1> bNþalMhat;Tak;TgnwgrUbmnþTMnak;TMngkñúgRtIekaN
01> bgðajfaRklaépÞ ΔABC epÞogpÞat; )A2sinbB2sina(41S 22 += .
02> tag R CakaMrgVg;carwkeRkA ΔABC. Rsayfa³
abc
)cba(RCcotBcotAcot222 ++
=++
03> kñúg ΔABC, AM Caemdüan ehIy . Rsayfaα=BMACsinBsin2)CBsin(cot −
=α .
04> kñúg ΔABC, CM Caemdüan ehIy . β=α= MCB;MCA
a. Rsayfa βα
=sinsin
BsinAsin
b. KNnamuM A, B, C CaGnuKmn_ βα, . 05> ABCD CactuekaNcarwkkñúgrgVg; ehIy AB = a, BC = b, CD = c, DA = d.
Rsayfa³)cp)(bp()dp)(ap(
2Atan
−−−−
= Edl 2
dcbap +++= .
06> RsayfaRKb; ΔABC eK)an³ 02Bcot)ac(
2Acot)cb(
2Ccot)ba( =−+−+− .
07> RsaybBa¢ak;faebIemdüan AA’ nig BB’ én ΔABC EkgKña eK)an³ )BcotA(cot2Ccot +=
11
2> bNþalMhat;Tak;TgvismPaB 08> RsaybBa¢ak;faRKb;ctuekaNABCD eK)anvismPaB . BD+AC<DC+AB
09> kñúgctuekaNe)a:g ABCD man O CacMnucRbsBVénGgát;RTUg ehIy S CaRklaépÞ. RsaybBa¢ak;fa ³ ¬smPaBekIteLIgenAeBlNa¦ . S2ODOCOBOA 2222 ≥+++
10> eKeGay ΔABC manRklaépÞesμ I 1 nigrgVas;RCug a, b, c (a ≥ b ≥ c) . RsaybBa¢ak;fa 2b ≥ .
11> eKeGay ΔABC manRklaépÞ S . ctuekaNEkg MNPQ carwkkñúg ΔABC . . tag S' CaRklaépÞ MNPQ .
Rsayfa S ≥ 2 . ( ) ( ) (BC∈Q&P,AC∈N,AB∈M )
'S
12> ABCD CakaerÉktamanRCugesμ I ! . M CacMnucmYyenAelIRCug [AD] & N Ca cMnucmYyenAelIRCug [CD] Edl . Rsayfa º45=NBM 2 -1 ≤ SMBN ≤
21 .
13> eKeGayctuekaN ABCD; O CacMnucRbsBVGgát;RTUg. tag S1=SAOB; S2=SCOD
S=SABCD. Rsayfa SSS 21 ≤+ .
14> eKeGayctuekaN ABCD manRklaépÞ S . Rsayfa S ≤ 81
(AC+BD)2 .
15> eKeGay ΔABC manmuMTaMgbICamuMRsYc. H CaGrtUsg;én ΔABC ehIy AM; BN;
CL CakMBs; . RsaybBa¢ak;fa³
12
a. 1=CLHL
+BNHN
+AMHM
b. 9HLCL
HNBN
HMAM
≥++
c. 4
BCHM.AM2
≤
16> eKeGay ΔABC & O CacMnucmYysßitkñúgRtIekaN. bnÞat; OA; OB; OC kat; BC; CA; AB erogKñaRtg; P; Q; R . RsaybBa¢ak;fa³
a. 1=CROR
+BQOQ
+APOP
b. 9ORCR
OQBQ
OPAP
≥++
17> eKeGaykaer QPSR carwkkñúg ΔABC EkgRtg; A Edl P; R enAelIRCug AB &
AC ehIy Q & R enAelIRCug BC. RsaybBa¢ak;fa BC ≥ 3QR
etIsmPaBekItenAeBlNa ?
18> eKeGayctuekaNe)a:g ABCD. Rsayfa AB.CD+AD.BC ≥ AC.BD .
19> eKeGay ΔABC; O CacMnucmYyenAkñúgRtIekaN. AO; BO; CO kat; BC; CA &
AB Rtg; P; Q; R. Rsayfa 23OROC
OQOB
OPOA
≥++ .
13
20> tamcMnuc O enAkñúg ΔABC KUsbnÞat;bIRsberog KañnwgRCug (AB); (BC); (CA) dUcrUbxagsþaM. RsaybBa¢ak;fa
23cba
≥δ
+β
+α
. bMrab; a; b; c
CaRklaépÞRtIekaN ehIy α; β; δ CaRklaépÞ ctuekaN.
21> eKeGayRtIekaN ABC man AM Caemdüan.
1. ebI A <2π Rsayfa³ a. BC2 < AB2 + AC2
b. BC < 2AM
2. ebI A >2A Rsayfa³ a. BC2 > AB2 + AC2
b. BC > 2AM
22> eKeGayRtIekaN ABC EkgRtg; A. D CaeCIgkMBs;KUsecjBI A. yk O1; O2
Cap©itrgVg;carwkkñúg ΔABD & ΔACD. bnÞat;kat;tam O1; O2 kat; (AB) & (AC)
Rtg; E & F. RsaybBa¢ak;fa 2SAEF ≤ SABC.
23> eKeGayRtIekaN ABC manmuMTaMgGs;CamuMRsYc. H CaGrtUsg;énRtIekaN nig a, b, c CargVas;RCug. Rsayfa (AH+BH+CH)2 ≤ a2 + b2 + c2.
14
24> eKeGayRtIekaN ABC carwkkñúgrgVg;p©it O. knøHbnÞat;BuHkñúgKUsecjBI A kat; BC Rtg; A1 nigrgVg;Rtg; A2. dUcKñacMeBaHknøHbnÞat;BuHKUsecjBI B & C kat;Rtg; BB1; B2; C1; C2. Rsayfa
43
BCACCC
CBABBB
CABAAAS
22
21
22
21
22
21 ≥+
++
++
= .
25> eKeGayctuekaN ABCD man AB=a; CD=c; AD=BC; . M, N, P, Q CacMnuckNþalerogKñaén AB, AC, CD, BD. RsaybBa¢ak;fa
º90BCDCDA =+
( )8
caS2
MNPQ−
≥ .
26> eKeGaymanrgVas;RCug a; b; c & knøHbnÞat;BuHkñúgénmuMTaMgbImanRbEvg la; lb; lc. Rsayfa
cba l1
l1
l1
c1
b1
a1
++<++ .
27> eKeGay ΔABC man A > B > C. O CacMnucmYyenAkñúgRtIekaN. bNþabnÞat; (AO); (BO); (CO) kat; BC; CA; AB Rtg; P; Q; R. RsaybBa¢ak;fa OP + OQ + OR < BC .
28> eKeGay ΔABC man O zitenAkñúgRtIekaNenaH. bNþarbnÞat; AO; BO; CO kat; BC; CA; AB Rtg; D; E; F. RsaybBa¢ak;fa³
a. 2CFOC
BEOB
ADOA
=++
b. 6OFOC
OEOB
ODOA
≥++
15
29> eKeGay ΔABC manmMu B CamMuTal. enAelI (BC) eKedABIrcMnuc M; N Edl BM=CN. Rsayfa AB + AC > AM + AN .
30> kñúg ΔABC edA A1 elI BC; B1 elI CA; C1 elI AB. RsaybBa¢ak;faya:gehac Nas;manmYykñúgcMenam ΔAB1C1; ΔBC1A1; ΔCA1BB1 manRklaépÞtUcCagrWesμ I
41 énRklaépÞ ΔABC.
31> RsaybBa¢ak;faRKb; ΔABC eK)an³ 8R27m.m.m
2
cba ≤ .
32> RsaybBa¢ak;fa³ . 2ccbbaa prlrlrl ≤++
33> eKeGay P CacMnucmYyenAkñúg ΔABC. cMgayBI P eTAkMBUl A, B, C KW x, y, z nig cMgayBI P eTA RCug AB, BC, CA KW p, q, r. RsaybBa¢ak;fa³ a. ( )rqp2zyx ++≥++ b. pqr8xyz ≥
34> kMnt;tMél p tUcbMputEdleKGacTaj)an . )cba(pS 4442 ++≤
35> eKeGayRtIekaN ABC manRklaépÞ S nig ma, mb, mc CaemdüanKUsecjBIkMBUl TaMg 3. RsaybBa¢ak;fa³ S33mmm 2
c2b
2a ≥++ .
36> RsaybBa¢ak;fa³ 32
cba abc)cba(rR4
r1
r1
r1
++≥++ .
37> eKeGayRtIekaN ABC manRklaépÞ S nig manRbEvgRCug a, b, c . RsaybBa¢ak;fa³ . 2222222 S16accbba ≥++
16
38> RsaybBa¢ak;fa³ ⎟⎠⎞
⎜⎝⎛ ++≥
−+
−+
− c1
b1
a12
cp1
bp1
ap1 .
39> eKeGayRtIekaN ABC Edlman . RsaybBa¢ak;fa³
CBA ≥≥
b
c
c
a
a
b
a
c
c
b
b
a
hh
hh
hh
hh
hh
hh
++≥++
40> RsaybBa¢ak;fa³ r
rRhm
hm
hm
c
c
b
b
a
a +≤++ .
41> RsaybBa¢ak;fa³ ( )( ) 22c
2b
2a
2c
2b
2a S27hhhmmm ≥++++ .
42> eKeGay ΔABCmanmMuRsYcbI.RsaybBa¢ak;fa: k. h a +h +h ≥ 9r b c
x. am
1 +bm
1 +cm
1≥
r2
sBaØa (=) ekItmanenAeBlNa ?
43> eKeGay ΔABCmankMBUl C.eKyk )1k(kABAC
≠= . KUsknøHbnÞat;BuH CM ,
AN nig BP. RsaybBa¢ak;fa :
k ) 2
MNP
ABC
k1k
SS
⎟⎟⎠
⎞⎜⎝
⎛+= x ) S MNP <
4SABC
44> eKeGayΔABC manTIRbCMuTMgn; G nig carwkkñúgrgVg;EdlmankaM R. bNþaemdüanEdlKUsecjBIkMBUl A , B , C kat;FñÚrgVg;erogKñaRtg; D , E , F. RsaybBa¢ak;fa: ⎟
⎠⎞
⎜⎝⎛ ++≤++≤
BC1
AC1
AB13
GF1
GE1
GD1
R3 .
17
45> eKeGayΔABC, tag D CacMnucenAelIRCug BC. enAelIbNþaRCug AB nig AC
eKedA cMnuc P nig Q erogKña. bNþabnÞat;EdlKUsecjBI P nig Q Rsbnwg AD
ehIykat;RCug BC Rtg; N nig M. RsaybBa¢ak;fa : S ≤ max
. smPaBekItmaneT,IgenAeBlNa ? MNPQ
{ ACDABD S,S }
3> bNþalMhat;Tak;TgtMéltUcnigFMbMput 46> eKeGay ΔABC manrgVas;RCug a, b, c nigbrimaRt 10cm . KNnaRklaépÞFM
bMputén ΔABC rYckMnt;RbePTRtIekaNEdlmanRklaépÞFMbMputenH .
47> eKeGay ΔABC manmMubICamMuRsYc. BIcMnuc I mYyzitenAkñúg ΔABC KUs IH, IK,
IL Ekgnwg BC, CA, & AB . rkTItaMgcMnuc I EdleFVIeGay AK2 + BH2 + CK2
tUcbMput.
48> eKeGaykaer ABCD manrgVas;RCug a . H CacMnucmYyenAelI [AC]. E & F Ca cMenalEkgén H elI [AB] & [BC]. kMnt;TItaMg H edIm,IeGay SDEF mantMél tUcbMput .
49> eKeGayRtIekaNsm½gS ABC; cMnuc M & N enAelI [AB] & [AC] Edl 1=
NCAN
+MBAM . rkTItaMgcMnuc M & N edIm,IeGay SAMN FMbMput.
18
50> eKeGay ΔABC man ∠A=30º, AB=c, AC=b, nig emdüan AM. KUsbnÞat;(d)mYy kat;tamTIRbCMuTMgn; Gén ΔABC ehIykat; ABRtg; Pnigkat; ACRtg; Q.
k. RsaybBa¢ak;fa: 3AQAC
APAB
=+
x. tag AP=x, rktMélGtibrmanigtMélGb,brmaén x.
51> eKeGaykaer ABCDmanRbEvgRCug a . tag M,N,P
CabIcMnucerogKñaenAelIbNþaRCug BC, CD, DA EdlΔMNP CaRtIekaNsmS½g. k) bgajfa : CN -AP = 2 DP.BM . 2 2
x) kMnt;TItaMg M,N,PedIm,IeGayΔMNPmanRklaépÞtUcbMput. 4> bNþalMhat;cMruH 52> enAelIRCug AC én ΔABC edAcMnuc E. tam E KUs (DE)//(BC) & (EF)//(AB)
. Rsayfa EFCADEBDEF S.S2S = .
53> tMbn; ABCDEFGHIJ pÞúkkaercMnYn 13 b:un²Kña ehIy tMbn;enHcarwkkñúgctuekaNEkg PQRS(emIlrUb).
eKeGay PQ = 28 & QR = 26 . KNnaRklaépÞtMbn;enH.
19
54> kñúgrUbxagsþaM ΔABC CaRtIekaNsmg½S ehIy P CacMnuc mYykñúgRtIekaN. bnÞat;Ekg PR, PS & PT RtUv)anKUs ecjBIcMnuc P eTAkan;RCugTaMgbIénRtIekaN. bgðajfa plbUkRklaépÞqñÚtTaMgbIesI μ
21 énRklaépÞ ΔABC.
55> tamcMnuc P kñúg ΔABC bnÞat; AX, BY, CZ RtUv)an KUsdUc)anbgðaj. bgðajfaebIRtIekaNEdlmanépÞqñÚt TaMgbImanRklaépÞesμ IKña enaHRtIekaNtUcTaMg6man RklaépÞesμ IKña .
56> kñúgrUbxagsþaM ABCD & AXYZ CaRbelLÚRkam Edl X enAelI [BC] nig D enAelI [YZ] . bgðajfa RbelLÚRkamTaMgBIrmanRklaépÞesμ IKña.
57> kñúgrUbxagsþaM bnÞat;QrmanRbEvg 5cm EkgbnÞat;edkEdl manRbEvg 310 cm
Rtg;cMnuckNþal ehIyFñÚekagCaEpñkmYy énrgVg;. KNnaRklaépÞEdlx½NÐedayFñÚekag nig bnÞat;edk .
58> tamcMnucmYykñúg ΔABC eKKUsbnÞat;ecjBIkMBUl RtIekaNkat;tamcMnucenH ehIyEck ΔABC Ca 6 Epñk . bYncMEnkmanRklaépÞ dUc)anbgðaj. KNna SABC.
20
59> ΔABC man AC = 7; BC = 24; º . M CacMnuckNþal [AB]; D enAEt mçagén(AB) CamYy C nig DA = DB = 15 . rkRklaépÞ ΔCDM .
90=C
60> eKeGay ΔABC manRklaépÞ ! Ékta. X, Y, Z CacMnucsßitenAelIGgát; [AB],
[BC], [AC] erogKañEdl 31
=CACZ
=CBBY
=ABAX . KNnaRklaépÞ ΔXYZ .
61> ΔABC CaRtIekaNEdleKeGaymun. eRbobeFobRklaépÞqekaNTaMgBIr.
62> [BC] CaGgát;p©iténrgVg;p©it O . ACacMnucmYyenAelIrgVg;EdlmuM . [EF] CaGgát;FñÚ nigCaemdüaT½rén [AO] . DCacMnuckNþalénFñÚtUc AB . bnÞat;kat;tam O ehIyRsb [AD] kat; [AC] Rtg; J . bgðajfa J Cap©iténrgVg; carwkkñúg ΔCEF .
º60>COA
63> eKeGayRbelLÚRkam ABCD EdlmankMBUl A CamMuRsYc. elIknøHbnÞat; AB &
CB tagcMnuc H & K Edl CH=CB & AK=AB . RsaybBa¢ak;fa³ a. DH=DK
b. ΔDKH ≅ ΔABK .
64> sg;kaer BRSC & DCTU enAelIRCugBIrénRbelLÚRkam ABCD. Rsayfa AC=ST ehIybnøayén (AC) Ekgnwg (ST).
21
65> eKeGaybnÞat; (XY) nigrgVg;p©it O ¬bnÞat;minkat;tamrgVg;eT¦. BIcMnuc A enAelI (XY) sg;bnÞat;b:HrgVg; (AB) & (AC) Rtg; B & C. RsaybBa¢ak;fa BC kat;tamcMnucnwgmYykalNa A rt;enAelI (XY).
66> eKeGay E; F; G CacMnucenAelIRCug AB; AC; BC én ΔABC Edl AG; BF &
CE kat;KñaRtg;cMnucmYy. Rsayfa 1FACF
GCBG
EBAE
=⋅⋅ .
67> tag P CacMnucenAelIrgVg;carwkeRkA ΔABC. A1; B1; C1 CaeCIgcMenalEkgKUsBI P
eTAelIRCug AB; BC; CA. Rsayfa A1; B1; C1 rt;Rtg;Kña.
68> eKeGayctuekaNEkg ABCD; M CacMnuckNþal [AB]; N CacMnuckNþal [CD];
Q enAelI [AC]. bnÞat; (QM) kat;bnÞat; (BC) Rtg; P. RsaybBa¢ak;fa . PNMMNQ =
69> kñúgrUbxagsþaM bnÞat; AB, CD & PQ CabnÞat;b:H rYmeTAnwgrgVg;TaMgBIrEdl A & C enAelIrgVg;Et mYy ehIy B & D enAelIrgVg;mYyeTot. cMnuc P
& D enAelI AB & CD . Rsayfa PB = QC.
70> bNþacMnuc P, Q, & R RtUv)anedAenAelIRCug BC, AC , AB erogKña én ΔABC
Edl AB31AR,CA
31CQ,BC
31BP === . bNþacMnuc X, Y RtUv)anedA
enAelIGgát; PR & QP Edl PQ31QX&PR
31PX == .Rsayfa (XY)//(BC).
22
71> eKeGay ΔABC mankMBUlRtg; A . tag I Cap©itrgVg;carwkeRkAΔABC, D CacMnuc kNþalRCug AB , E CaTIRbCMuTMgn;én ΔACD . RsaybBa¢ak;fa : IE CD⊥ .
72> eKeGayctuekaNe)a:g ABCD, enAelIbNþaGgát; AB, BC , CD , DA eKedAbNþa cMnuc M ,N , P , Q Edl AQ = DP = CN = BM .RsaybBa¢ak;fa ebI MNPQ
CakaerenaH ABCDk¾CakaerEdr.
73> eKeGayctuekaN ABCD carwkkñúgrgVg;p©it(O),Ggát;RTUgTaMgBIrénctuekaNkat;Kña Rtg;M. BIcMnuc P enAelIRCug AB KUsbnÞat; PM , BIcMnuc Q enAelIRCug BC
KUsbnÞat; QM. RsaybBa¢ak;faebI PM CD⊥ enaH QM AD⊥ Edr.
74> rgVg; ¬S¦ kaM R b:HnwgbnÞat;BIrRsbKña (t1) & (t2) . rgVg; (S1) kaM r1 b:HnwgrgVg; ¬S¦ nigbnÞat; (t1) . rgVg; (S2) kaM r2 b:HnwgrgVg; ¬S¦ nigbnÞat; (t2) ehIyrgVg; (S1) &
(S2) CargVg;b:HKñapgEdr . KNna R CaGnuKmn_én r1 & r2 .
75> ABCD CactuekaNe)a:gEdl AB = 8 ; BC = 6 ; BD = 10 ; &
. KNna CD . D=A
C=DBA
76> kñúg ΔABC ; . D CacMnucmYyenAelI [AC] Edl . bgðajfa AB + DB = BC .
ACABCAB == º&100ˆ
DBC=DBA
77> kñúg ΔABC edAcMnuc P mYy . enAelIRCug AC & BC edAcMnuc M & L Edl . RsaybBa¢ak;faebI D CacMnuckNþal [AB]
enaH DM = DL . º90CMPCLP&CBPCAP ===
23
78> kñúg ΔABC eKman Edl . Rsayfa BM=AC .
(AC∈M;º50=B;º30=A ) CB=CM
79> kñúgRtIekaNsm)at ABC EdlmanmMukMBUl edAcMnuc M Edl
. KNna .
º80=C
º10=BAM;º30=ABM CMA
80> eKeGayctuekaNBañy ABCD man)at AB=a; CD=b & RCugeRTt AD=c;
BC=d & Ggát;RTUg AC=p; BD=q . Rsayfa p2 + q2 = c2 + d2 + 2ab .
81> D CacMnucmYyenAelIRCug [AB] én ΔABC Edl AB=4AD. P CacMnucmYyenA elIrgVg;carwkeRkA ΔABC Edl . bgðajfa PB=2PD . BCA=PDA
82> eKeGay ΔABC sm½gScarwkkñúgrgVg;. P CacMnucmYyenAelIFñÚ BC. Rsayfa PA = PB + PC .
83> eKeGaykaer ABCD mYy. edAcMnuc P kñúgkaerEdl . Rsayfa PCD CaRtIekaNsm½gS.
º15=ABP=BAP
84> kñúgrUbxagsþaM ABCD CabnÞat;Rtg;Edl AB=BC=CD=2 & AF=DE=2; BE=4 &
FC=CE. KNna FB.
85> eKeGayrgVg;p©it (O) kaM r b:HrgVg; (O’) kaM R. bnÞat; (L) b:HrgVg;TaMgBIrRtg; S &
T. KNna |ST| .
24
86> eKkMnt; |PQR| CaRklaépÞén PQR. bNþaGgát;RTUgén ABCD kat;KañRtg;E. ]bmafa |AEB|=3; |DEC|=10; |BEC|=2|AED|. KNna |AED|.
87> eKeGay ΔABC dUcrUbxageRkam.eday (SR)//(CB);(TU)//(AC);(PQ)//(BC) cUrrkRklaépÞ ΔABC edaydwgfabnÞat; (PQ); (TU); (SR) Eck ΔABC CaBIr cMEnkmanRklaépÞesμ IrKañ ehIy SXYZ=1.
88> eKeGay ABCD CactuekaNe)a:gman K; L; M; N CacMnuckNþalerogKñaén DC;
DA; AB; BC. ]bmafa AK kat; BL; DN Rtg; P; S ehIy CM kat; BL; DN
Rtg; Q; R. KNnaRklaépÞéncTuekaN PQRS edaydwgfa SABCD=3000;
SAMQP=513 nig SCKSR=388 .
89> eKeGayctuekaN ABCD man AD= 3 ; ; E & F
Cap©itrgVg;carwkkñúgRtIekaN ABD & ACD; ebI EF =
º60DCADBA ==
213 − . KNna BC.
90> enAelIRCugEkg [AC] & [BC] én ΔABC sg;xageRkAnUvkaer ACKL &
BCMN. (BL) kat; (AC) & (AN) erogKñaRtg; P & R. (AN) kat; (BC) Rtg; Q. Rsayfa SCPRQ = SABR.
91> bnÞat; (L) kat;RCug [AB] & [AD] énRbelLÚRkam ABCD Rtg; E & F erogKña. ]bmafa G CacMnucRbsBVrvag (L) & (AC). Rsayfa
AFAD
AEAB
AGAC
+= .
25
92> eKeGayrgVg; (O1; R1); (O2; R2) b:HKña ehIyrgVg; (O3; R3) b:HeTAnwgrgVg; O1; O2
rYcb:HeTAnwgbnÞat;b:HeRkArvag O1 & O2. RsaybBa¢ak;fa 213 R
1R1
R1
+= .
93> kñúgbBa©ekaN ABCD manrgVas;RCug 1; 2; 3; 4 & 5 edayminKitBIlMdab;. eKyk F; G; H; I CacMnuckNþalerogKñaén AB; BC; CD; DE. X CacMnuckNþalén
[FH] & Y CacMnuckNþalén [GI]. Ggát; [XY] mantMélRbEvgCacMnYnKt;. KNnatMélEdlGacmanrbs; AE.
94> enAelIRCug BC én ΔABC edAcMnuc P Edl PC = 2BP. KNna ebI .
BCA
º60CPA;º45CBA ==
95> bNþaGgát;RTUg AC & CE énqekaNniy½t ABCDEF RtUvEckedaycMnuc M & N Edl λ== CE:CNAC:AM . KNna λ ebI B, M, N rt;Rtg;Kña.
96> rUbxagsþaMekIteLIgBIRtIekaNsm)atb:unKñacMnYn 6 EdlRtUv)an tMerobedaymineGayCan;Kña. RtIekaNnImYy²man)atRbEvg 1Ékta nig RCugRbEvg 2Ékta. KNnaRbEvg AB .
97> enAEpñkxagkñúgctuekaN ABCD edA M Edl ABMD
CaRbelLÚRkam. Rsayfa ebI enaH . MBCMDC = MCBDCA =
98> kñúgrUbxagsþaM rgVg;BIrb:HKña ehIycarwkkñúgknøHrgVg;Edl mankaM 2cm. ebIrgVg;FMb:HGgát;p©iténknøHrgVg;Rtg;p©it C
rbs;va KNnakaMénrgVg;tUc.
26
99> kñúgrUbxagsþaM D CacMnuckNþalFñÚ AC nig DE ⊥ AB . Rsayfa AE = EB + BC .
100> ΔABC EkgRtg; A manRbEvgbNþaRCug BC=a , AB=c , CA=b
carwkkñúgrgVg;Ggát; p©it BC.cMnuc P mYyenAelIrgVg;Edl Pnig A enAsgxag
BC.tam P KUs PK ⊥ BC Rtg; K , PL ⊥ AC Rtg; L nig PM ⊥ AB Rtg; M.tagRbEvgbNþaGgát; PK, PL nig PM CalMdab;erogKña x ,y ,z. rktMélGb,brmaénplbUk : S =
zc
yb
xa
++ .
101> eKeGayctuekaNe)a:g ABCD carwkkñúgrgVg;p©it O . eKdwgfaknøHbnÞat;BuHén bNþamMu BAD nig ABC kat;KñaRtg;cMnuc E enAelIRCug CD. k ) bgðajfa : AD + BC = CD .
x ) eKdwgfa 1kCBCD
>= . KNna BCE
ADE
SS .
102> eKeGay ΔABC manRCug a; b; c & kMBs; ha; hb; hc. RsaybBa¢ak;faebI
)ocp(p1
)bp(p1
)ap(p1
h1
h1
h1
cba −+
−+
−=++
enaH ΔABC CaRtIekaNsm½gS.
103> eKeGay ΔABC man 0 < B < 90º. tag AH; AD; AM CakMBs;; CaknøHbnÞat;BuH nigCaemdüanKUsecjBI A. Rsayfa D enAcenøaH M & H.
27
104> kñúg ΔABC KUskMBs; AD, BE & CF. Rsayfa Rr
P'P= Edl P' CabrimaRt
ΔEDF & P CabrimaRt ΔABC.
105> RsaybBa¢ak;faebIRtIekaN ABC manRCugEdlepÞogpÞat;³ enaH ΔABC CaRtIekaNsm½gS .
)ba(c)ac(b)cb(a)cba(2 222222333 +++++=++
106> RsaybBa¢ak;faebI 2R9mmm cba =++ enaH ΔABC CaRtIekaNsm½gS .
107> RsayfaRKb; ΔABC eK)an³ 0
2Bcot)ac(
2Acot)cb(
2Ccot)ba( =−+−+− .
108> kMnt;RbePTRtIekaN ABC EdlepÞogpÞat;³
a. )cba)(cba(41S +−−+=
b. 2)cba(36
3S ++=
109> BIp©it O énrgVg;carwkeRkA ΔABC KUsbnÞat;Ekg OA', OB' & OC' eTAelIRCug BC, CA & AB erogKña(muM ΔABC CamuMRsYc) . Rsayfa rR'OC'OB'OA +=++
28
5> bNþalMhat;Rblg Olympiad 110> P CacMnuckñúg ΔABC. PA kat; BC Rtg; D; PB kat; AC Rtg; E & PC kat;
AB Rtg; F. bgðajfay:agehacNas;manmYykñúgcMenam PFCP;
PEBP;
PDAP man
tMélminelIsBI 2 nigy:agehacNas;manmYymantMélFMCag 2.
111> kñúgrgVg;mYyedAGgát;FñÚ AB nig cMnuc C edAelIrgVg;Edl cMgayBI C eTA AB esμ I 4cm. ebI CA=6cm & CB=10cm KNnaGgát;p©iténrgVg;.
112> eKeGayctuekaNe)a:g ABCD EdlGgát;RTUg AC & BD kat;KñaRtg; O . ]bmafa CakaMrgVg;carwkkñúgénRtIekaN OAB, OBD, OCD &
ODA erogKña ehIyepÞogpÞat; 4321 r,r,r,r
4231 r1
r1
r1
r1
+=+ . RsaybBa¢ak;fa ABCD
CactuekaNcarwkeRkArgVg;.
113> kñúg ΔABC eKeGay D, E, F CacMnuckNþalénRCug BC, CA, AB nig P, Q, R
Ca cMnuckNþalénemdüan AD, BE, CF erogKña. RsayfatMélpleFob T
xageRkam minGaRs½ynwgrUbragrbs;RtIekaN ehIyrktMélenaH.
222
222222
CABCABCQCPBRBPARAQT
+++++++
=
114> kñúg ΔPQR man PQ=8, QR=13 & PR=15. bgðajfamancMnuc S mYyenAelI PR ¬EtminenAcMnuccugénGgát;eT¦ Edl PS & QS mantMélKt;.
29
115> eKeGaybBa©ekaNniy½t ABCDE carwkkñúgrgVg; (O). enAelIFñÚ AB én (O)
edAcMnuc M ( M minsßitelI A rW B eLIy) . RsaybBa¢ak;fa³ MEMCMDMBMA +=++
116> eKeGayRtIekaN ABC mYy. sg;RtIkaN ABR, BCP, CAQ EdlsßitenACab; xageRkA ΔABC ehIyman nig
. Rsayfa . º30ACQPCB,º45QACCBP ====
º15RABRBA == RPQR&º90PRQ ==
117> eKeGay a, b, c CaRCugén ΔABC nig A CaRklaépÞ. RsaybBa¢ak;fa A34cba 222 ≥++ . etIsmPaBekIteLIgeBlNa ?
118> kaMénrgVg;carwkeRkARtIekaNsm)atmYymanRbEvg R nig kaMrgVg;carwkkñúgén RtIekaNenHKW r. bgðajfacMgayrvagp©iténrgVg;TaMgBIrKW )r2R(R − .
119> RtIekaN ABC manRbErgRCug a, b, c. bnÞat;bIEdlb:HeTAnwgrgVg;carwkkñúg RtUv)anKUsRsbeTAnwgRCugTaMgbI. bnÞat;b:HnImYy²pÁMúCamYyRCugBIr²énRtIekaN begáIt)anCaRtIekaNEdlmancMnYnsrub3. KNnaplbUksrubrvagRklaépÞrgVg; carwkkñúgRtIekaNTaMg3 nig rgVg;carwkkñúg ΔABC.
30
CMBUk 3 dMeNaHRsay
01> bgðajfa )A2sinbB2sina(41S 22 +=
tamRTwsþIbT sin ³ BsinR2b,AsinR2a == eK)an³ )A2sinBsinB2sinA(sinR)A2sinbB2sina(
41S 22222 +=+=
)BAsin(BsinAsinR2)AcosBsinBcosA(sinBsinAsinR2 22 +=+=
Bit CsinBsinAsinR2)Cº180sin(BsinAsinR2 22 =−=
dUcenH )A2sinbB2sina(41S 22 += .
02> Rsayfa
abc)cba(RCcotBcotA
222 ++=++cot
eyIgman abc
)acb(R
R2abc2
acb
AsinAcosAcot
222
222
−+=
−+
==
dUcKñaEdreK)an abc
)bca(RBcot222 −+
= , abc
)cab(RCcot222 −+
=
dUcenHeK)an abc
)cba(RCcotBcotAcot222 ++
=++ .
03> Rsayfa
CsinBsin2)CBsin(cot −
=α tamRTwsþIbT sin eyIg)an ³
31
)Bsin(2
aMABsin
BMsin
cα+
==α
⇒ Csin2
)CBsin(Csin2
Asinc2
asin
)Bsin( +===
αα+
⇔ Csin2
BcosCsinCcosBsinsin
BcossincosBsin +=
αα+α
⇔ Bcoscsin2
BcosCsinCcosBsincot.Bsin −+
=α
⇒ Csin2
)CBsin(cotBsin −=α
dUcenH CsinBsin2)CBsin(cot −
=α .
04> a. Rsayfa =
sinsin
BA α
βsinsin
tamRTwsþIbT sin kñúg ΔACM & ΔBCM eyIg)an³ )2(
BsinCM
sinBM),1(
AsinCM
sinAM
=β
=α
eday AM = BM
tam (1) & (2) eK)an BsinAsin
sinsin
=βα
b. KNnamuM A, B, C CaGnuKmn_ βα, π ebI enaH β=α α=α−== 2C&2
BA
ebI , ]bmafa β≠α β>α enaH β>α sinsin tam (a) :
BsinAsin
sinsin
=βα ⇔
BsinAsinbsinsin
BsinAsinsinsin
Bsinsin
Asinsin
++α
=−
β−α=
β=
α
32
⇒β+αβ−α
=+−
sinsinsinsin
BsinAsinBsinAsin
⇔2
cot2
tan2
BAcot2
BAtan β+αβ−α=
+−
eday enaH C=β+α2
tan2
BAcot222
BA β+α=
+⇒
β+α−
π=
+
⇒2
cot2
tan2
tan2
BAtan β+αβ−α=
β+α−
⇔2
cot2
tan2
BAtan 2 β+αβ−α=
−
tag 2
cot2
tan2
tan 2 β+αβ−α=
ϕ enaH 2
tan2
BAtan ϕ=
−
⇒
⎪⎪⎩
⎪⎪⎨
⎧
β+α−
π=
+
ϕ=
−
222BA
22BA
⇒
⎪⎪⎩
⎪⎪⎨
⎧
ϕ−
β+α−
π=
ϕ+
β+α−
π=
222B
222A
.
05> Rsayfa³
)cp)(bp()dp)(ap(
2Atan
−−−−
=
eyIgman Acos1Acos1
2Acos2Asin
2Atan
2
2
2
+−
==
eday CcosAcosº180CA −=⇒=+
tamRTwsþIbTkUsIunIskñúg ΔABD & ΔCBD: Acosbc2cbCcosbc2cbAcosad2daBD 2222222 ++=−+=−+=
⇒)bcad(2
cbdaAcos2222
+−−+
=
)bcad(2cbda1Acos1
2222
+−−+
−=−•
33
)bcad(2
cbbc2daad2 2222
++++−−
=
)bcad(2
)da()cb( 22
+−−+
=
)bcad(2
)dacb)(adcb(+
−++−++=
bcad
)dp)(ap(2+
−−=
)bcad(2)cb()da(
)bcad(2cbda1Acos1
222222
+−−+
=+
−−++=+•
bcad
)cp)(bp(2+
−−=
eK)an )cp)(bp()dp)(ap(
coA1Acos1
2Atan2
−−−−
=+−
=
dUcenH )cp)(bp()dp)(ap(
2Atan
−−−−
= .
06> Rsayfa 0
2Bcot)ac(
2Acot)cb(
2Ccot)ba( =−+−+−
eyIgman 2Bcos
2Csin
2Ccos
2Bsin
2Csin
2Bsina
2C
2Bsin
2Csin
2Bsina
2Acos
2Csin
2Bsina
r+
=⎟⎠⎞
⎜⎝⎛ +
==
⇒ )2Ccot
2B(cotra +=
dUcKñaEdr eK)an ⎟⎠⎞
⎜⎝⎛ +=
2Ccot
2Acotrb
⎟⎠⎞
⎜⎝⎛ +=
2Acot
2Bcotrc
34
⎟⎠⎞
⎜⎝⎛ −=−•
2Acot
2Bcotr)ba(
⎟⎠⎞
⎜⎝⎛ −=−•
2Bcot
2Ccotr)cb(
⎟⎠⎞
⎜⎝⎛ −=−•
2Ccot
2Acotr)ac(
eK)an 02Bcot)ac(
2Acot)cb(
2Ccot)ba( =−+−+− .
07> Rsayfa )BcotA(cot2Ccot += yk G CaTIRbCMuTMgn;én ΔABC eK)an
94m
32AG m2
a2
a2 =⎟⎠
⎞⎜⎝⎛=
)ac2b2(91 222 −+=
dUcKñaEdr eK)an )bc2a2(91BG 2222 −+=
eday AA’ ⊥ BB’ ⇒ 222 BGAGAB +=
⇔ )c4ba(91c 2222 ++=
⇔ 222 c5ba =+ ⇔ 2222 c4cba =−+ ⇔ 2c4Ccosab2 =
tamRTwsþIbT sin eK)an CsinR2c,BsinR2b,AsinR2a === ⇒ 2)CsinR2(4Ccos.BsinR2.AsinR2.2 =
⇔ (1) Csin2CcosBsinAsin 2=
eyIgBinitü BcosAsin
)BAsin(2CsinCcos)BcotA(cot2Ccot +=⇔+=
35
(2) Csin2CsinBsinAsin 2=⇔
tam (1) & (2) eK)an )BcotA(cot2Ccot += . 08> Rsayfa BD+AC<DC+AB
tag O CaGgát;RTUgctuekaN ABCD eyIgman AC=OA+OC
BD=OB+OD ⇒ AC + BD = (OA+OB)+(OC+OD) > AB + DC dUcenH BD+AC<DC+AB . 09> RsaybBa¢ak;fa³ S2ODOCOBOA 2222 ≥+++
eyIgman S = S1 +S2 +S3 + S4
2S = 2S1 +2S2 +2S3 + 2S4
=21 .2OA.ODsinα1+ 2
1 .2 OA.OBsinα2
+21 .2 OB.OC sinα3+ 2
1 .2 OC.OD sinα4
=OA.ODsinα1 + OA.OBsinα2 + OB.OC sinα3 + OC.OD sinα4
eday sinα1 ≤ 1; sinα2 ≤ 1; sinα3 ≤ 1; sinα4 ≤ 1 ⇒ OA.ODsinα1 + OA.OBsinα2 + OB.OCsinα3 + OC.OD sinα4
≤ OA.OD + OA.OB + OB.OC + OC.OD
≤ 2
ODOC OCOB OBOA ODOA 22222222 +++++++
= OA2+OB2+OC2+OD2
dUcenH 2S ≤ OA2+OB2+OC2+OD2
36
smPaBekIteLIgkalNa OA=OB=OC=OD; α1=α2=α3=α4=90º KW ABCD
Cakaer.
10> RsaybBa¢ak;fa 2b ≥ eyIgman S=
21 bcsinA ≤
21 bc (sinA ≤ 1)
eday b ≥ c ⇒ 21 bc ≤
21 b2
⇒ S ≤ 21 b2 dUcenH 2b ≥ .
11> Rsayfa ' S2S≥
tag H CakMBs; ΔABC KUsecjBInMBUl A. eyIgman ;BCAH
21S ⋅⋅= NPMN'S ⋅=
eday (MN) // (BC) ⇒BCMN
ACAN
= (1)
(PN) // (AH) ⇒ AHPN
CACN
= (2)
tam (1) & (2) eK)an 2ACCNAN
AHBCPNMN ⋅
=⋅⋅
⇔ 2ACCNAN
S2'S ⋅=
tam Cauchy: CNAN4)CNAN(CNAN2CNAN 222 ⋅≥+⇒⋅⋅≥+
⇔ CNAN4
AC2
⋅≥
eK)an ⇒≤ 2
2
AC4AC
S2'S 'S2S≥ .
37
-1 ≤ SMBN ≤ 21
12> Rsayfa 2
eyIgman BNBM42º45sinBNBM
21SMBN ⋅⋅=⋅⋅=
tag )SSS(SSS 321ABCMBN ++−==
⎟⎠⎞
⎜⎝⎛ ⋅⋅+⋅⋅+⋅⋅−= DNDM
21NCBC
21ABAM
211
)DNDMNCAM(211 ⋅++−=
[ ]DN).AM1(NCAM211 −++⋅−=
[ ]DNNCDNAMAM211 ++⋅−−=
[ ]CD)DN1(AM211 +−−=
[ ] S21CN.AM1CN.AM211S −=⇒+−=
müa:geTot )CN1)(AM1(42BNBM
42S 22 ++=⋅⋅=
1CNAMCNAM42 2222 +++⋅=
⇔ [ ]11BN1BM)S21(81S 2222 +−+−+−=
⇔ 1BNBMS4S41S8 2222 −+++−=
⇔ S2
8BNBM42
28BN.BM2BNBMS4S4 222 =⋅⋅⋅=≥+=+
⇔ 0S2S2S2 2 ≥−+ ⇔ 0)22S2(S ≥−+ eday S > 0 ⇒ 12S021S −≥⇔≥−+
ehIy )1CN.AM(211S +−=
38
ebI M RtYtelI A rW D ⇒21
211S0CN.AM =−=⇔=
ebI M minRtYtelI A rW D ⇒21
211S0CN.AM =−<⇔>
⇒ 21S ≤
dUcenH 21S12 MBN ≤≤− .
13> Rsayfa SSS 21 ≤+
eyIgman α⋅⋅= sinOBOA21
1S
α⋅⋅= sinOCOD21S2
α⋅⋅×α⋅⋅= sinOCOD21sinOBOA
21S.S 21
Et β=α⇒=β+α sinsinº180 eK)an β⋅⋅×β⋅⋅= sinOCOD
21sinOBOA
21S.S 21
naMeGay OBCOAD21 S.SS.S =
müa:geTot OBCOAD21OBCOAD21 SS2SSSSSSS ⋅++≥+++= ⇔ ( )2212121 SSSS2SSS +=⋅++≥ dUcenH 21 SSS +≥ .
14> Rsayfa S ≤
81
(AC+BD)2
eyIgman OADODCOBCAOB SSSSS +++=
39
Et )OB.OA(21BOAsinOBOA
21SAOB ≤⋅=
eFVIdUcKñaEdr eK)an )OC.OB(21SOBC ≤
)OC.OD(21SOCD ≤
)OD.OA(21SOAD ≤
⇒ )OA.ODOD.OCOC.OBOA.OB(21S +++≤
⇔ [ ])OCOA(OD)OCAO(OB21S +++≤
⇔ DB.AC21S ≤
tam Cauchy ⇒ DB.AC2DBAC 22 ≥+
⇔ DB.AC4
)DBAC( 2
≥+
enaH 2)DBAC(81S +≤ .
15> RsaybBa¢ak;fa³ a. 1=
CLHL
+BNHN
+AMHM eyIgman BC.AM
21SABC =
BC.HM21SHBC =
AMHM
SS
ABC
HBC =⇒
40
dUcKñaEdreK)an ;BNHN
SS
ABC
HAC = CLHL
SS
ABC
HAB =
enaH 1SS
SSSS
CLHL
BNHN
AMHM
ABC
ABC
ABC
HABHACHBC ==++
=++
b. 9HLCL
HNBN
HMAM
≥++
tamsMrayxagelI ⇒ ⎟⎠
⎞⎜⎝
⎛++=++
AHBHACHBCABC S
1S
1S
1SHLCL
HNBN
HMAM
⎟⎠
⎞⎜⎝
⎛++++=
AHBHACHBCAHBHACHBC S
1S
1S
1)SSS(
tam Cauchy: 3AHBHACHBCAHBHACHBC SSS3SSS ⋅⋅≥++
3
AHBHACHBCAHBHACHBC SSS13
S1
S1
S1
⋅⋅≥++
⇒ 9HLCL
HNBN
HMAM
≥++
c. 4
BCHM.AM2
≤
ΔABM & ΔHMC CaRtIekaNEkg (mMuRCugEkgerogKña) BCLMAB =
⇒ ΔABM ∼ ΔHMC enaH MC.BMHM.AMHMMC
BMAM
=⇔=
tam Cauchy: MC.BM2MCBM 22 ≥+
⇔ MC.BM4
)MCBM( 2
≥+ Et BM+MC=BC
dUcenH 4
BCHM.AM2
≤ .
41
16> RsaybBa¢ak;fa³ a. 1=
CROR
+BQOQ
+APOP
tag [AH] CakMBs; ΔABC
H´ CacMenalEkgBI O→AH
⇒ [HH´] CakMBs; ΔOBC
BC'.HH
21S
BC.AH21S
OBC
ABC
=
= ⇒
AH'HH
SS
ABC
OBC =
eday (OH´) // (BC) ⇒APOP
AH'HH=
⇒ APOP
SS
ABC
OBC =
dUcKñaEdreK)an CROR
SS;
BQOQ
SS
ABC
OAB
ABC
OAC ==
eK)an 1S
SSSCROR
BQOQ
APOP
ABC
OABOACOBC =++
=++
b. 9ORCR
OQBQ
OPAP
≥++
⎟⎠
⎞⎜⎝
⎛++=++
OABOACOBCABC S
1S
1S
1SORCR
OQBQ
OPAP
( ) 9S
1S
1S
1SSSOABOACOBC
OABOACOBC ≥⎟⎠
⎞⎜⎝
⎛++++=
17> RsaybBa¢ak;fa BC ≥ 3QR
eyIgman³
42
BCSQPB = ¬mMumanRCugEkgerogKña¦ ⇒ ΔPBQ & ΔSCR CaRtIekaNekgman³
BCSQPB = enaH ΔPBQ ∼ ΔSCR
vi)ak RC.BQPD.SRRCPQ
SRBQ
=⇔= (1)
Et RC.BQQRQRSRPQ 2 =⇒==
QR3QRRC.BQ2QRRCBQBC =+≥++= smPaBekIteLIgkalNa RCBQ = tam (1) ⇒ enaH QPBQQRBQ 22 =⇒= º45CBA =
dUcenH smPaBekIteLIgkalNa ABC CaRtIekaNEkgsm)at. QR3BC ≥
18> Rsayfa AB.CD+AD.BC ≥ AC.BD
tag K CacMnucmYyenAkñúgctuekaNEdl CDBKAB;CBDKBA ==
eK)an ΔAKB ∼ ΔBCD
vi)ak BDAB
CBBK
DCAK
== enaH (1) AK.BDDC.AB =
müa:geTot CBDDBKCBK;DBKKBADBA +=+=
enaH EtCBKDBA =CBBK
BDAB
= ⇒ ΔABD ∼ ΔBKC
vi)ak CKAD
BCBD
BKAB
== ⇒ CK.BDAD.BC = (2)
43
yk (1)+(2)⇒ AC.BD)CKAK(BDAD.BCDC.AB ≥+=+ (AK+KC≥AC)
dUcenH AB.CD+AD.BC ≥ AC.BD
smPaBekIteLIgkalNa K elI (AC) . 19> Rsayfa 23
OROC
OQOB
OPOA
≥++
tagS SS;SS;SS;S ABC3OAB2OCA1OBC = = == 321 SSSS ++=⇒
eyIgman1
32
1
321
1 SSS1
SSSS
SS +
+=++
= (1)
Et OPOA1
OPOPOA
OPAP
SS
1
+=+
== (2)
(1) & (2) eK)an 1
32
1
32
SSS
OPOA
SSS
OPOA +
=⇔+
=
eFVIdUcKñaEdreK)an 2
31
SSS
OQOB +
= ; 3
21
SSS
OROC +
=
tam Cauchy: 3232 SS2SS ≥+ ⇒ ( )23232 SS)SS(2 +≥+
2
SSSS 32
32
+≥+⇔
naMeGay ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+≥
1
3
1
2
1
32
SS
SS
21
S.2SS
OPOA
26
SS
SS
SS
SS
SS
SS
21
OROC
OQOB
OPOA
3
2
3
1
2
2
2
1
1
3
1
2 ≥⎥⎥⎦
⎤
⎢⎢⎣
⎡+++++≥++
44
dUcenH 23OROC
OQOB
OPOA
≥++ .
20> RsaybBa¢ak;fa
23cb
≥δ
+β
+aα
tam Cauchy: 3abc3cbaαβδ
≥δ
+β
+α
eday 2sinOF.OE21a α=
3sinOH.OG21b α=
1sinOI.OJ21c α=
321 sinsinsin
OJ.OI.OH.OG.OF.OE81abc
ααα×
=⇒
müa:geTot 2sinOI.OH α=α 3sinOE.OJ α=β 1sinOG.OF α=δ ⇒ 321 sinsinsin.OJ.OI.OH.OG.OF.OE ααα=αβδ
naMeGay 81abc
=αβδ
enaH 23cba
≥δ
+β
+α
.
21. sMrayvismPaB 1.eyIgman : AcosAC.AB2ACABBC 222 −+=
ebI ⇒>⇒π
< 0Acos2
A 222 ACABBC +<
45
müa:geTot : ( ) 222222 BCBC2BCACAB2AM4 −>−+= ⇒>⇒ 22 BCAM4 BCAM2 >
2.RsaydUcsMnYrTI (1) Edr. 22. RsaybBa¢ak;fa 2SAEF ≤ SABC
tagrgVg; mankaM 1O 1R
rgVg; mankaM nig AD = h 2O 2R
eyIgman : IJJOIO 22 −=
Et HDADAHAMJO2 −=== 22 RhPOAD −=−= 11 RLOIJ ==
enaH 212 RRhIO −−= eFVIdUcKñaEdrcMeBaH 211 RRhIO −−= dUcenH . 21 IOIO =
man 21OIO⊥Δ 21 IOIO = enaH 21OIOΔ CaRtIekaNEkgsm)at. Et ehIy º45AFOIOO 221 == º90OMF 2 =
naMeGay CaRtIekaNEkgsm)at. FMO2Δ
vi)ak 22 RMFMO == hRRhMFAHMFAMAF 22 =+−=+=+= Δ AEF CaRtIekaNEkgman hAFAEº45EFA ==⇒=
46
21
ACABAC.AB
BCh.BC
BCh
SS
BC.h21S
h21S
222ABC
AEF
ABC
2AEF
≤+
===⇒
=
=
dUcenH ABCAEF SS2 ≤ . 23. Rsayfa (AH+BH+CH)2 ≤ a2 + b2 + c2 eday CamMurYmrvag 1BAA∠ 1AHC⊥Δ & BAA1⊥Δ
⇒ ∼ 1AHCΔ BAA1Δ
vi)ak AB.ACAA.AHABAH
AAAC
111
1 =⇔=
kñúg AC.AcosACACACAcos:CAC 1
11 =⇒=Δ
enaH 2
acbAcosAC.ABAA.AH222
1−+
==
eFVIdUcKñaEdr 2
bcaBB.BH;2
cbaCC.CH222
1
222
1−+
=−+
=⇒
naMeGay2
cbaCC.CHBB.BHAA.AH222
111++
=++
müa:geTot tamlMhat;elx 15>a : 1CCHC
BBHB
AAHA
1
1
1
1
1
1 =++
2CCCH
BBBH
AAAH1
CCCHCC
BBBHBB
AAAHAA
1111
1
1
1
1
1 =++⇒=−
+−
+−
⇔
tamvismPaB Buniakovsky :
47
2
11
11
11
CC.CHCCCHBBBH
BBBHAA.AH
AAAH
⎟⎟⎠
⎞⎜⎜⎝
⎛++
( )111111
CC.CHBB.BHAA.AHCCCH
BBBH
AAAH
++⎟⎠
⎞⎜⎝
⎛++≤
( )
( ) 2222
2222
cbaCHBHAH2
cba2CHBHAH
++≤++⇒
++≤++⇔
dUcenH ( ) 2222 cbaCHBHAH ++≤++ . 24. Rsayfa
43
BCACCC
CBABBB
CABAAA
22
21
22
21
22
21 ≥+
++
++
=S
eyIgman : CaknøHbnÞat;BuHmuM A [ )2AA CABA 22 =⇒
Et AC.BACA.ABBC.AA 222 +=
naMeGay ( )ACABCABC.AA 22 +=
⇔ 2
2
AACA
ACABBC
=+
nig man : 2ACAΔ CAA 21Δ
CAAAABACA 2221 ==
CamuMrYm 2A∠
naMeGay CAA~ACA 212 ΔΔ vi)ak
2
2
2
21
AACA
CAAA
=
enaH ( )ACAB2BC
CABAAA
ACABBC
CAAA
22
21
2
21
+=
+⇔
+=
48
eFVIdUcKñaEdr ( ) ( )BCAC2AB
BCACCC;
BCAB2AC
CBABBB
22
21
22
21
+=
++=
+⇒
⎥⎦⎤
⎢⎣⎡
++
++
+=
BCACAB
BCABAC
ACABBC
21S
( ) ⎥⎦
⎤⎢⎣
⎡−⎟⎠
⎞⎜⎝⎛
++
++
+++= 3
BCAC1
BCAB1
ACAB1ACBCAB
21
( ) ⎥⎦
⎤⎢⎣
⎡−⎟⎠
⎞⎜⎝⎛
++
++
++++++= 6
BCAC1
ACAB1
ACAB1BCACBCABACAB
41
43
≥
dUcenH 43
BCACCC
CBABBB
CABAAAS
22
21
22
21
22
21 ≥+
++
++
= .
)(25> RsaybBa¢ak;fa
8caS
2
MNPQ−
≥ tag CacMnucRbsBVrvagbnøay(DA)&(BC) 'A
eday º90BCDCDA =+
º90CAD ' =⇒
M kNþal AB
Q kNþal DB
( ) ( ) ( )1AD//MQ&2
ADMQ =⇒
N kNþal AC
P kNþal DC ( ) ( ) (2AD//NP;2
ADNP =⇒ )
tam (1)nig (2)naMeGay MNPQ CaRbelLÚRkam.
49
Et ( )CA//MN;DA//MQº90CADNMQ;MQ2
AD2
BCMN ''' ===== ⇒ MNPQ Cakaer.
2QNMQS
22
MNPQ ==
tag K kNþal [AD]2c
2DCKN;
2a
2ABKQ ====⇒
Et 2
caKQKNQN
−=−≥
eyIg)an ( )8
caS2
MNPQ−
≥ .
26> Rsayfa
cba l1
l1
l1
c1
b1
a1
++<++
tag [ )'AA CaknøHbnÞat;BuHmuM A. tam B KUsbnÞat;Rsb ' kat;bnøay (CA) Rtg; D. AA
enaH ( ) ( ) 'AABABD;CA'AADB'AA//BD ==⇒
Et CA'A'AAB =
ADABABDADB =⇒=⇒
müa:geTot ( ) ( )DB.ACADAC
DB.ACDC
'AA1
DBA'A
DCAC'AA//BD +
==⇔=⇒
⇒ ( ) ⎟⎠⎞
⎜⎝⎛ +=⎟
⎠⎞
⎜⎝⎛ +=
+=
++
>c1
b1
21
AB1
AC1
21
AB.AC2ABAC
ADAB.ACADAC
'AA1
( )1c1
b1
21
l1
a⎟⎠⎞
⎜⎝⎛ +>⇒
50
RsaydUcKñaEdr ( ) ( )3c1
b1
21
l1;2
c1
a1
21
l1
cb⎟⎠⎞
⎜⎝⎛ +>⎟⎠
⎞⎜⎝⎛ +>⇒
bUkGgÁnwgGgÁén ( ) ( ) ( )c1
b1
a1
l1
l1
l13;2;1
cba++>++⇒
dUcenH c1
b1
a1
l1
l1
l1
cba++>++ .
27 . RsaybBa¢ak;fa OP + OQ + OR < BC
eyIg]bmafa º90BPA >
APAB
PBABPA>>⇒
Et ABBCCA >⇒>
enaH BC > AP
eFVIdUcKñaEdr BQBC;CRBC >>⇒
eyIgman CROR
SS;
BQOQ
SS;
APOP
SS
ABC
OAB
ABC
OAC
ABC
OBC ===
enaH 1CROR
BQOQ
APOP
=++
BCOROQOP
1BCOR
BCOQ
BCOP
<++⇒
<++⇔
dUcenH BCOROQOP <++ .
28> RsaybBa¢ak;fa³
51
a. 2CFOC
BEOB
ADOA
=++
eyIgman ADOD1
ADODAD
ADOA
−=−
=
BEOE1
BEOB
−= ; CFOF1
CFOC
−=
eK)an =++CFOC
BEOB
ADOA
213CFOF
BEOE
ADOD3 =−=⎟⎠
⎞⎜⎝⎛ ++−=
dUcenH 2CFOC
BEOB
ADOA
=++ .
b. 6OFOC
OEOB
ODOA
≥++
eyIgman 1SS1
ODAD
ODODAD
ODOA
OBC
ABC −=−=−
=
dUcKñaEdr ⇒ 1SS
OFOC;1
SS
OEOB
OAB
ABC
OAC
ABC −=−=
⇒ 3S
1S
1S
1SOFOC
OEOB
ODOA
OABOACOBCABC −⎟
⎠
⎞⎜⎝
⎛++=++
3S
1S
1S
1)SSS(OABOACOBC
OABOACOBC −⎟⎠
⎞⎜⎝
⎛++++=
639 =−≥
52
29> Rsayfa AB + AC > AM + AN
bnøay AM eGay)an AN = MD
eyIgman NAMNMACNA +=
⇒ DMBNMACNA => ⇒ BDAC >
ΔABD man ACABBDABAD +<+< ⇔ ACABMDAM +<+ ⇔ ACABANAM +<+ dUcenH AB + AC > AM + AN . 30> sMraybBa©ak; tag ABCBCA3ABC2BAC1 SS;SS;SS;SS
111111====
eK)an AC.ABAB.AC
SS 111 = ;
BC.ABBA.BC
SS 112 = ;
AC.BC
CB.CASS 113 =
⇒ 211
211
211
3321
BCCA.BA.
ACCB.AB.
ABBC.AC
SS.S.S
=
Et 41
AB4AB
AB4)BCAC(
ABBC.AC
2
2
2
211
211 ==
+≤
⇒ 41.
41.
41
SS
.SS.
SS 321 ≤
⇒ ya:gehacNas;manmYykñúgcMenam SS
;SS;
SS 321 mantMéltUcCagrWesμ I
41
⇒ S41S1 ≤ rW S
41S2 ≤ rW S
41S3 ≤
dUcenH ya:gehacNas;manmYykñúgcMenam ΔAB1C1; ΔBC1A1; ΔCA1BB1
53
manRklaépÞtUcCagrWes μ I 41 énRklaépÞ ΔABC .
31> RsaybBa¢ak;fa 8R27m.m.m
2
cba ≤
eyIgman 2
am2cb2
2a
22 +=+ ⇒ 2222
a ac2b2m4 −+=
⇒ )CsinBsinA(sinR4.3)cba(3)mmm(4 22222222c
2b
2a ++=++=++
eyIgBinitü CsinBsinAsin 222 ++
CcosBcosAcos22 += [ ])BAcos()BAcos()BA(cos2 2 −+++−=
49)BA(sin
41)BAcos(
21)BAcos(
49 2
2
≤⎥⎥⎦
⎤
⎢⎢⎣
⎡−+⎥⎦
⎤⎢⎣⎡ −++−=
eK)an 3 2c
2b
2a
2c
2b
2a
2
m.m.m3mmm4R27
≥++≥
⇒ 8R27m.m.m
2
cba ≤ .
32> RsaybBa¢ak;fa³ 2
ccbbaa prlrlrl ≤++
ap)cp)(bp)(ap(p
r)cp)(bp)(ap(p)ap(r aa −−−−
=⇒−−−=−=S eyIgman
ehIy cb
)ap(p.bc2la +
−=
eK)an ap
)cp)(bp)(ap(p.
cb)ap(p.bc2
r.l aa −−−−
+−
=
54
cb
)cp)(bp(p.bc2+
−−=
2a.p)
2cpbp(p)cp)(bp(p =
−+−≤−−≤
dUcKñaEdr eK)an 2c.pr.l;
2b.pr.l ccbb ≤≤
2ccbbaa p
2cbapr.lr.lr.l =⎟⎠⎞
⎜⎝⎛ ++
≤++⇒ .
33> RsaybBa¢ak;fa³ a. ( )rqp2zyx ++≥++
tag P' CacMnucqøúHén P eFobknøHbnÞat;BuHmMu [AX) rbs;mMu A .
eK)an xA'PPA == p'K'PPA == r'L'PPL == eyIgman CA'PBC'PAB'PABC SSSS ++=
⇔ )'H'P'h('ahbpcraha =++= ⇔ ax)'hh(abpcr a ≤−=+
⇒ pabr
acx +≥ (1)
eFVIdUcKñaEdr eK)an qcbr
cay +≥ (2)
pbaq
bcz +≥ (3)
yk (1) + (2) + (3) ⇒ qbc
cbp
ab
abr
ca
aczyx ⎟
⎠⎞
⎜⎝⎛ ++⎟
⎠⎞
⎜⎝⎛ ++⎟
⎠⎞
⎜⎝⎛ +≥++
tamvismPaB Cauchy ⇒ )qpr(2zyx ++≥++
b. pqr8xyz ≥
55
tam (1), (2) & (3) kñúgsMnYr (a) nig edayeRbIvismPaB Cauchy eK)an³
pqbacz,rq
cab2y,rp
abc2x 222 ≥≥≥
⇒ pqr8rqpcbacba8xyz 222
222
222
=≥ .
34> kMnt;tMél p tUcbMput
tamrUbmnþ Heron eyIg)an )cp)(bp)(ap(pS2 −−−=
tamrUbmnþ Cauchy eK)an 27p
3)cba(p3)cp)(bp)(ap(
33
=⎥⎦⎤
⎢⎣⎡ ++−
≤−−−
⇒ 44
2 )cba(2716
127pS ++
×=≤ (1)
tamvismPaB Bunyakovsky:
+ )cba(3)cba)(111()cba( 2222222 ++=++++≤++
⇒ 22224 )cba(9)cba( ++≤++
+ )cba(3)cba( 4442222 ++≤++
eK)an (2) )cba(27)cba( 4444 ++≤++
tam (1) & (2) ⇒ )cba(27.2716
1S 4442 ++×
≤
⇔ )cba(161S 4442 ++≤
ebI 161p ≥ enaH )cba(pS 4442 ++≤
dUcenH tMél p tUcbMputKW 161p = .
56
35> RsaybBa¢ak;fa³ S33mmm 2
c2b
2a ≥++
eyIgman
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+=+
+=+
+=+
2cm2ba
2bm2ac
2am2cb
22c
22
22b
22
22a
22
⇒ )cba(43mmm 2222
c2b
2a ++=++ (1)
tamrUbmnþ Heron: )cp)(bp)(ap(pS −−−=
tamrUbmnþ Cauchy eK)an 27p
3)cba(p3)cp)(bp)(ap(
33
=⎥⎦⎤
⎢⎣⎡ ++−
≤−−−
⇒33
pS2
≤ ⇔ 33.4
)cba(333.4
)cba)(111(33.4
)cba(S2222222 ++
=++++
≤++
≤
⇔ 4
)cba(3S33222 ++
≤ (2)
tam (1) & (2) ⇒ S33mmm 2c
2b
2a ≥++ .
36> RsaybBa¢ak;fa³ 3
2
cba abc)cba(rR4
r1
r1
r1
++≥++
eyIgnman cba r)cp(r)bp(r)ap(S −=−=−=
eyIg)an S2
cbaSp
S)cp()bp()ap(
r1
r1
r1
cba
++==
−+−+−=++
tamvismPaB Cauchy eyIg)an 33
cba Sabc
23
r1
r1
r1
≥++
57
eday prR4
abcS == ⇒ abcpr
R23)pr.(
R16cba
abc23
r1
r1
r1 2
32
222cba
=
⎟⎟⎠
⎞⎜⎜⎝
⎛≥++
⇒ 32
cba abc)cba(rR4
r1
r1
r1
++≥++ .
37> RsaybBa¢ak;fa³ 2222222 S16accbba ≥++
eyIman³ + ⇔ AsincbS4 2222 = Asincb4S16 2222 = + Acoscb4)acb(ca2ba2cb2cba 2222222222222444 =−+=−−+++
⇒ 222222224442 cb4ca2ba2cb2cbaS16 =−−++++⇔ (1) )cba()cabacb(2S16 4442222222 ++−++=
tamvismPaB Bunyakovsky:
)cba()abc)(cab()cabacb( 444444444222222 ++=++++≤++ (2)
tam (1) & (2) ⇒ 2222222 S16accbba ≥++ .
38> RsaybBa¢ak;fa³ ⎟⎠⎞
⎜⎝⎛ ++≥
−+
−+
− c1
b1
a12
cp1
bp1
ap1
eyIgman c4
2bpap
c)bp)(ap(
cbp
1ap
12 =
⎟⎠⎞
⎜⎝⎛ −+−
≥−−
=−
+−
(1)
RsaydUcKñaEdr eyIg)an a4
cp1
bp1
≥−
+−
(2) ; b4
ap1
cp1
≥−
+−
(3)
yk (1)+(2)+(3) ⇒ ⎟⎠⎞
⎜⎝⎛ ++≥
−+
−+
− c1
b1
a12
cp1
bp1
ap1 .
58
39> RsaybBa¢ak;fa³
b
c
c
a
a
b
a
c
c
b
b
a
hh
hh
hh
hh
hh
hh
++≥++
eyIgman cba chbhahS2 === ⇒
c
a
b
c
a
b
hh
ac;
hh
cb;
hh
ba
===
eyIgRsayfa b
c
c
a
a
b
a
c
c
b
b
a
hh
hh
hh
hh
hh
hh
++≥++
⇔ cb
ac
ba
ca
bc
ab
++≥++
⇔ 222222 abbccabaaccb ++≥++ ⇔ 0)ac)(cb)(ab( ≥−−− (*)
tamsm μtikm μ ⇒ ⇒ (*) Bit CBA ≥≥ cba ≥≥
dUcenH b
c
c
a
a
b
a
c
c
b
b
a
hh
hh
hh
hh
hh
hh
++≥++ .
40> RsaybBa¢ak;fa³
rrR
hm
hm
hm
c
c
b
b
a
a +≤++
tag (O, R) CargVg;carwke®kA ΔABC. M, N, P CacMnuckNþal BC, CA & AB erogKña. eK)an OMOAAM +< ⇔ OMRma +<
⇔ aaa
a
hOM
hR
hm
+<
naMeGay cbacbac
c
b
b
a
a
hOP
hON
hOM
h1
h1
h1R
hm
hm
hm
+++⎟⎠
⎞⎜⎝
⎛++<++
59
eyIgBinitü + 1S2S2
S2cOPbONaOM
hOP
hON
hOM
cba==
++=++
+ r1
Sp
S2cba
h1
h1
h1
cba==
++=++
⇒ 1rR
hm
hm
hm
c
c
b
b
a
a +<++ ⇔ r
rRhm
hm
hm
c
c
b
b
a
a +≤++ .
41> RsaybBa¢ak;fa³ ( )( ) 22
c2b
2a
2c
2b
2a S27hhhmmm ≥++++
eyIgman
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+=+
+=+
+=+
2cm2ba
2bm2ac
2am2cb
22c
22
22b
22
22a
22
⇒ 3 2222222c
2b
2a cba
49)cba(
43mmm ≥++=++ (1)
müa:geTot 3222
2222
22c
2b
2a cba
1S3.4c1
b1
a1S4hhh ≥⎟⎠
⎞⎜⎝⎛ ++=++ (2)
yk ⇒ )2()1( × ( )( ) 22
c2b
2a
2c
2b
2a S27hhhmmm ≥++++ .
42> k ) Rsayfa h +h +h ≥ 9r a b c eyIgman : ( ) ⎟⎠
⎞⎜⎝⎛ ++=⇒++==
ac
ab1rhcbarS2ah aa
eFVIdUcKñaEdr : ⎟⎠⎞
⎜⎝⎛ ++=⎟⎠
⎞⎜⎝⎛ ++=
cb
ca1rh,
bc
ba1rh cb
60
⎟⎠⎞
⎜⎝⎛ ++++++=++⇒
cb
bc
ca
ac
ba
ab3rhhh cba
tamvismPaB Cauchy : 6cb
bc
ca
ac
ba
ab
≥+++++
r9hhh cba ≥++⇒
x ) Rsayfa am
1 +bm
1 +cm
1 ≥ r2
eyIgman : AMOMAO ≥+ OMAMAO −≥⇒
AMOM1
AMAO
−≥⇒
eday AM
1AH1AMAH ≥⇒≤
AHOM1
AMAO
−≥⇒
S
S1mR
SS1
AMAO BOC
aABC
BOC −≥⇒−≥⇔
eFVIdUcKñaEdreK)an : S
S1mR,
SS1
mR AOB
c
COA
b−≥−≥
R2
m1
m1
m12
mR
mR
mR
cbacba≥++⇔≥++⇒ .
43. k ) Rsayfa 2
MNP
ABC
k1k
SS
⎟⎠
⎞⎜⎝⎛
+=
tag : 3BMN2AMP1CPNABC SS,SS,SS,SS ==== 321MNP SSSSS −−−=⇒
61
eday AN CaknøHbnÞat;BuH k
ABAC
NBNC
==⇒
1kBNCB
+=⇒ nig 1k
kBCCN
+=
eyIgman : Δ CPN ∼ Δ ABC
22
1PNC
1kk
BCCN
SS
SS
⎟⎠⎞
⎜⎝⎛
+=⎟⎠
⎞⎜⎝⎛==⇒
S.1k
kS2
1 ⎟⎠⎞
⎜⎝⎛
+=⇒
eyIgman : 1k
1BCBN
SS
MBC
MBN
+==
Δ APM ≅ Δ BMN (C-m-C)
1k
SSS1k
1S
SSBMNAPM
BMNAPM
+=+⇒
+=
+⇒
( ) S.1k
kS.1k
k1k
SSS 2
2
MNP +=⎟⎠
⎞⎜⎝⎛
+−
+−=⇒
( ) 22
MNP
ABC
k1k
k1k
SS
⎟⎠⎞
⎜⎝⎛
+=+
=⇒ .
x ) Rsayfa S < MNP 4SABC
tamvismPaB Cauchy eyIg)an : 2k
1k ≥+
sBaØa (=) ekItmankalNa : k = 1
EtebI k ≠ 1 ⇒k
1k + > 2 ⇒2
k1k ⎟⎠
⎞⎜⎝
⎛+ > 4
62
dUcenH : MNP
ABC
SS > 4 ⇒ < MNPS
4SABC .
44 . RsaybBa¢ak;fa: ⎟
⎠⎞
⎜⎝⎛ ++≤++≤
BC1
AC1
AB13
GF1
GE1
GD1
R3
tag AM, BN, CP CabNþaemdüanén ΔABC,
tag amAM,aBC,bAC,cAB ==== eyIg)an :
4am.MDMC.MBMA.MD
2
a =⇒=
naMeGay a
2
m4aMD = eday GD = GM + MD =
3
am4a.
3m2
m4am
31
a
2a
a
2
a =≥+
eyIg)an : .BC
3a3
GD1
=≤
RsaydUcKñaEdrcMeBaHGgát; GE nig GF,ehIybUkGgÁnigGgÁeyIg)an : ⎟
⎠⎞
⎜⎝⎛ ++≤++
AB1
AC1
BC13
GF1
GE1
GD1
Et ,m32
GA a= naMeGay .a3m4
m8
m4am
31
m32
GDGA
22a
2a
a
2
a
a
+=
+=
Gnuvtþn_tamrUbmnþKNnaemdüaneyIg)an : ( ) 2222a acb2m4 −+=
enaH ( )222
222
2222
222
cbaac2b2
a3ac2b2ac2b22
GDGA
++−+
=+−+−+
=
eFVIdUcKñaEdrcMeBaHpleFob : GFGC;
GEGB eyIg)an :
63
.3cba
c3b3a3GFGC
GEGB
GDGA
222
222
=++++
=++
eyIgman : ,GDAG1
GDGDAG
GDAD
+=+
= ehIyeFVIdUcKñaEdrcMeBaH GFCF,
GEBE
eyIg)an : ( )16GFGC
GEGB
GBGA3
GFCF
GEBE
GDAD
=+++=++
edaybNþaGgát; AD, BE, CF FMCag 2R naMeGaytam (1)eyIg)an : ⎟
⎠⎞
⎜⎝⎛ ++=++≤++=
GF1
GE1
GD1R2
GFR2
GER2
GDR2
GFCF
GEBE
GDAD6
naMeGay : .R3
R26
GF1
GE1
GD1
=≥++
dUcenH ⎟⎠⎞
⎜⎝⎛ ++≤++≤
AB1
AC1
BC13
GF1
GE1
GD1
R3 .
45 . RsaybBa¢ak;fa : S ≤ max MNPQ { }ACDABD S,S tag x
ABAP
= nig ,yACAQ
= cMeBaH 0 < x, y < 1,eyIg)an : ,xy
AC.ABAQ.AP
SS
ABC
APQ == naMeGay ABCAPQ S.xyS = (1)
tamrUbmnþeyIgman : x1BABP
BDBN
−==
y1CACQ
CDCM
−==
naMeGay ( ) ABD2
BNP S.x1S −= (2) ( ) ACD
2CMQ S.y1S −= (3)
tam (1) , (2) nig (3) eyIg)an : CMQBNPAPQABCMNPQ SSSSS −−−=
64
= ( ) ( )[ ] ( ) ( )[ ] ACD2
ABD2 S.y1xy1S.x1xy1 −−−+−−−
= ( ) ( ) ACD2
ABD2 S.yxyy2S.xxyx2 −−+−−
eday ( ) 0xy2xxxyx2 2 >−−=−− nig ( ) 0yx2yyxyy2 2 >−−=−− naMeGay ( ) ( )[ ] { }ACDABD
22MNPQ S,Smax.yxyy2xxyx2S −−+−−≤
( ) ( )[ ] { }( )[ ] { }ACDABD
2MNPQ
ACDABD2
MNPQ
S,Smax.1yx1S
S,Smax.yxyx2S
−+−≤⇔
+−+≤⇔
naMeGay { }ACDABDMNPQ S,SmaxS ≤ sBaØa ( = ) ekItmankalNa ACDABD SS = nig
x + y = 1,ehIy BD = AC nig 1ACAQ
ABAP
=+ . 46> KNnaRklaépÞFMbMputén ΔABC
tag S CaRklaépÞRtIekaN ABC
⇒ )cp)(bp)(ap(pS −−−= Et 52
10p == enaH )c5)(b5)(a5(5S −−−= ⇔ S2 = )c5)(b5)(a5(5 −−−
tamvismPaB Cauchy: )c5()b5()a5( −+−+− ≥ 3 )c5)(b5)(a5(3 −−− ⇔ 15-(a+b+c) ≥ 3 )c5)(b5)(a5(3 −−− (a + b + c = 10)
⇔ 3
35⎟⎠⎞
⎜⎝⎛ ≥ )c5)(b5)(a5( −−− ⇔ )c5)(b5)(a5(5
35
3
4
−−−≥ = S2
⇒ S ≤ 9
325 dUcenH MaxS = 9
325
smPaBekIteLIgkalNa a = b = c ⇒ ΔABC CaRtIekaNsm½gS.
65
47> kMnt;TItaMgcMnuc I
eyIgman³ 222222 ILIKAKILAIAL −+=−= 222222 IHILBLIHBIBH −+=−= 222222 IKIHCHIKICCK −+=−=⇒ 222222 AKCHBLCKBHAL ++=++⇔ )KCAK()CHBH()BLAL()CKBHAL(2 222222222 +++++=++
yk a; b CaBIrcMnYnviC¢man enaHtamvismPaB Cauchy eK)an ab2ba 22 ≥+
⇔ 2
)ba(ba2
22 +≥+
eK)an³
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=+
≥+
=+
≥+
=+
≥+
2AC
2)KCAK(KCAK
2BC
2)HCBH(CHBH
2AB
2)BLAL(BLAL
2222
2222
2222
enaH )ACBCAB(21)CKBHAL(2 222222 ++≥++
)ACBCAB(41)CKBHAL( 222222 ++≥++
)ACBCAB(41)CKBHALmin( 222222 ++=++
smPaBekItmankalNa AL=LB; BH=HC; CK=KA
naMeGay L; H; K CacMnuckNþalerogKñaén AB; BC;AC
Et (IL) ⊥ (AB); (IH) ⊥ (BC); (IK) ⊥ (AC)
66
dUcenHedIm,IeGayplbUktUcbMputluHRtaEt I Cap©itrgVg;carwkeRkA ΔABC. 48> kMnt;TItaMg M edIm,IeGay SDEF mantMéltUcbMput
eyIgman³ )SSS(SS FCDAEDBEFABCDDEF ++−=
⎥⎦⎤
⎢⎣⎡ ++−−−= ay
21ax
21)ya)(xa(
21a2
( )ayaxxyayaxa21a 22 +++−−−=
( ) ⎥⎦
⎤⎢⎣
⎡ +−≥−=
4)yx(a
21xya
21 2
22
Δ⊥HFC man HFC CaRtIekaNEkgsm)at ⇒= º45FCH
⇒ HF = y = EB = a-x ⇒ x + y = a
naMeGay 8a3
4aa
21S
222
DEF =⎟⎠
⎞⎜⎝
⎛−≥ ⇒
8a3Smin
2
DEF =
smPaBekIteLIgkalNa x=y ⇔ x=a-x ⇔ 2ax =
eK)an E kNþal [AB]
dUcenH H kNþal [AC] eFVIeGay tUcbMput. BEFS
49> rkTItaMgcMnuc M & N edIm,IeGay SAMN FMbMput
eyIgman 1NCAN
MBAM
=+
⇔ 3NCAC
MBAB1
NCNCAC
MBMBAB
=+⇔=−
+−
67
⇔ 3ABNC1
MB1
=⎟⎠⎞
⎜⎝⎛ + 3
NC.MB)NCMB(AB=
+⇔
NC1
MB1
AB3
+=⇔
4)NCAB)(MBAB(
4)NCAC)(MBAB(º60sinAN.AM
21SAMN
−−=
−−==
4
NC.MB2AB4
NC.MBAB)NCMB(AB 22 −=
++−=
tam Cauchy:NC.MB
12AB3
NC.MB12
NC1
MB1
≥⇔≥+
9AB4NC.MB2
2
≥⇔
naMeGay 36
AB4
9AB4AB
S2
22
AMN =−
≤
tMél 36
ABSmax2
AMN =
tMélenHekIteLIgkalNa 3ABMB
1MB
1NCMB =⎟⎠⎞
⎜⎝⎛ +⇒=
3AB2MB =⇔ &
3AC2NC =
dUcenHtMél SAMN FMbMputkalNa 3AC2NC;
3AB2MB == .
50> k ) RsaybBa¢ak;fa: 3
AQAC
APAB
=+
KUs BE Rsbnwg PQ , KUs CF Rsbnwg PQ
CMFBME Δ≅Δ⇒ ( m-C-m ) vi)ak MFME =
eyIgman : AGAF
AQAC,
AGAE
APAB
==
68
( ) ( ) 323.2
AGAM2
AGMFAMMEAM
AGAFAE
AQAC
APAB
===++−
=+
=+⇒
3AQAC
APAB
=+⇒
x ) rktMélGtibrmanigtMélGb,brmaén x
kalNa (d)kat;tam B nig G ⇒cMnuc P ≡ B cxABAP =⇒=⇒ kalNa (d)kat;tam C nig G ⇒ P ≡ N
CacMnuckNþalén AB
cx2c
2cxANAP ≤≤⇒=⇒=⇒
dUcenHtMélGtibrmaén x KW c nig tMélGb,brmaén x KW 2c .
51. k ) bgðajfa : CN -AP = 2 DP.BM 2 2
eyIgman: ( ) 22222 CNBMaCNMCMN +−=+=
ehIy : ( ) ( ) 22222 APBMaAPBMABMP −+=−+=
( ) ( )2222 APBMaCNBMa −+=+−⇒ ( )APaBM2AP.BM2aBM2APCN 22 −=−=−⇔
DP.BM2=
dUcenH : . BM.DP2APCN 22 =−
x ) kMnt;TItaMg M,N,P
eyIgman : 4
3MPS2
MNP = , naMeGay : S Gb,brma ⇔ MP xøIbMput MNP
⇔ MP = a ⇔ MP // AB
69
eyIg)an : Δ PDN ≅ Δ MCN ⇒ ND = NC
⇒ N CacMnuckNþal CD naMeGay 2
3aDPCM == .
52> Rsayfa EFCADEBDEF S.S2S = eyIgman BDEFBDF S
21
=S
ADEF
Bsin.BF.ADBsin.BF.EF
Dsin.DE.AD.2/1Bsin.BF.BD.2/1
SS
S2S
ADE
BDF
ADE
BDEF
==
==⇒
eyIgBinitüeXIjfa ΔADE ∼ ΔEFC ⇒ ADE
EFC
SS
ADEF
=
⇒ ADEEFCBDEFADE
EFC
ADE
BDEF S.S2SSS
S2S
=⇒= .
53> KNnaRklaépÞtMbn; ABCBDEFGHIJ kñúgkaernImYy² KUsbnÞat;QrBIr nig edkBIrdUc)anbgðaj. Ggát;TaMg 4 pÁÚb
nwgRCugkaer begáIt)anRtIekaNEkg 4 b:unKña. yk a & b CaRbEvgRCugCab;én RtIekaNTaMgenaH. tamrUb eK)anRbErg beNþay nig TTwg énctuekaNEkgKW
. ba5&b3a5 ++
eK)an 1b&5a
26ba528b3a5
==⇒⎩⎨⎧
=+=+
70
RbEvgRCugénkaernImYy²KW 26ba 22 =+ RklaépÞénkaernImYy²KW 262626 =× dUcenHRklaépÞéntMbn; ABCBDEFGHIJ KW 3332613 =× . 54> bgðajfa plbUkRklaépÞqñÚtTaMgbIesI μ
21 énRklaépÞ ΔABC
KUsbnÞat; (UV), (WX), (YZ) RsbnwgRCugnWg RCugTaMg 3 én ΔABC .
enaH muMTaMg3én ΔPVY mantMél 60º ⇒ ΔPVY CaRtIekaNsm½gS ⇒ kMBs; PR Eck ΔPVY CaBIrb:unKñaEdlmçag
manépÞqñÚt. dUcKñaEdrcMeBaH ΔPXU & ΔPZW (1)
eyIgBinitüctuekaN PWBV man (WP)//(BV) & (BW)//(VP)
⇒ PWBV CaRbelLÚ®kam ⇒ BP Eck PWBV CaBIrb:unKñaEdlmçagmanépÞqñÚt dUcKñaEdrcMeBaH ctuekaN PXCY & PZAU (2)
tam (1) & (2) ⇒ plbUkRklaépÞqñÚtTaMgbIesI μ21 énRklaépÞ ΔABC.
55> RsayfaRtIekaNtUcTaMg6manRklaépÞes μ IKña edayeRCIserIsykÉktasmRsbmYy eyIgGac
sn μtfaRklaépÞénRtIekaNnImYy² EdlmanépÞqñÚt mantMélesμ I 1ÉktaRklaépÞ.
71
tag x, y, z CaRklaépÞén ΔBXP, ΔCYP & ΔAZP
eday ΔPBX & ΔPCX mankMBs;dUcKña ⇒
CXBXx
CXBX
1x
=⇔=
dUcKñaEdrcMeBaH ΔABX & ΔACX ⇒ CXBX
y2zx1=
+++
enaH y1z1x
y2zx1x
++
=⇔+++
= (1)
dUcKñaenHEdreK)an x1y1z,)2(
z1x1y
++
=++
= (3)
yk ⇒ )3()2()1( ×× 1xyz =
⇒ CarwsmYyénRbB½n§smIkar 1zyx ===
]bmafaeyIgGacrkrwsepSgeTot)an ebI tam (1) ⇒ 1x > 2)x1()z1)(y1(
z1x1
x1y1yz +>++⇒
++
>++
⇒> (a) 1xz)x1()z1( 22 >>⇒+>+⇒
Et x1y1xy1yxz1y +<+⇒<⇒<⇒=
tam (3) 1x1y1z <
++
= (b)
tam (a) & (b) ⇒ eFVIeGayRbB½n§smIkarKμanrws 1x >
⇒ x, y, z minGacFMCag 1
enaH tam (1) ⇒ 1x < yz < ⇒ z1x1
x1y1
++
<++
⇒ xz)x1()z1()x1()z1)(y1( 222 <⇒+<+⇒+<++
tam (2) 1z1z1
z1x1y >
++
>++
= minGac dUcenHRbB½n§smIkarmanrwsEtmYyKt;KW 1zyx ===
72
dUcenHRtIekaNtUcTaMg6manRklaépÞes μ IKña . 56> bgðajfa RbelLÚRkamTaMgBIrmanRklaépÞesμ IKña tag P CacMnucRbsBVrvagbnøay BC & ZY . eday (AX)//(DP) & (AD)//(XP)
naMeGay AXPD CaRbelLÚRkam eyIgeXIjfaRbelLÚRkam AXYZ & AXPD man RCug AX rYmKña nig ZY &
DP RtYtelIKña ⇒ (1) AXPDAXYZ SS =
dUcKñaEdrcMeBaH ABCD & AXPD ⇒ (2) AXPDABCD SS =
tam (1) & (2) ⇒ ABCDAXYZ SS = . 57> KNnaRklaépÞEdlx½NÐedayFñÚekag nig bnÞat;edk tag O Cap©itrgVg;kaM r EdlmanFñÚ AB
PX CabnÞat;Qr ⇒ OP & PX RtYtelIKña eyIgman 35BXAX,5PX === eK)an 5rOX −= AOX CaRtIekaNEkg ⇒ 222 r)35()5r( =+− ⇒ 10r =
naMeGay OB23BX = ⇒ º60BOX =
RklaépÞcMnitrgVg;KW 3
10010.3
2 π=
π
73
RklaépÞ ΔABO KW 325310.5.21
= ⇒ RklaépÞEdlx½NÐedayFñÚekag nig bnÞat;edkKW 325
3100
−π .
58> KNna SABC
tag AOC2OBA1 SS;SS ′′ == eyIgman BOAsinOBAO
21SAOB ⋅⋅⋅=
'AOBsin'OAOB21S AOB ⋅⋅⋅=′
eK)an 'AOBsin
BOAsin'OA
AOSS
'OBA
AOB ⋅=
eday 'AOBsinBOAsinº180'AOBBOA =⇒=+
enaH 235
3040'OA
OASS
'OBA
AOB =+
==
eFVIdUcKñaEdrcMeBaH ΔOAB; ΔOAB´⇒OB
'OB70S
SS 1
AOB
A'OB ==
ΔOCB; ΔOCB´⇒OB
'OB35S
84SS
2OCB
OC'B =+
=
ΔAOC; ΔOCA´⇒ 2'OA
AOS
84SSS
2
1
'OCA
AOC ==+
=
eK)an ⎪⎪⎩
⎪⎪⎨
⎧
=+
+=
)2(S2
84S
)1(35S
8470S
21
2
1
(1)⇒ 5880352
84SS5880)35S(S 1121 =⎟⎠
⎞⎜⎝⎛ +
+⇔=+
11760S70S84S 1121 =++⇔
74
133'
011760S154S 121
=Δ=−+⇔
eK)an S1 = 56; S2 = 70
dUcenH SABC= 70+56+84+40+30+35=315 ÉktaRklaépÞ . 59> KNnaRklaépÞ ΔCDM
kñúg ΔABC: 25247BCACAB 2222 =+=+=
eday AD=DB=15 ⇒ ADB CaRtIekaNsm)at Et M kNþal [AB] ⇒ [DM] CaemdüaT½r [AB] ⇒ (DM) ⊥ (MB) kñúg Δ⊥DMB:
1125
4115
22515MBDBDM 2
2222 ==−=−=
⇒ 611
215115
DBDMMBDsin =
⋅==
enaH 4
11125611
22515MBDsin.BM.DBSDMB =⋅⋅==
kñúg ΔABC: 257
ABBCABCsin ==
MBDCBACBD −= )MBDCBAsin(CBDsin −=
CBAcos.MBDsinMBDcos.CBAsinCBDsin −=⇔
CBAcos611MBDcos
257CBDsin ⋅−⋅=⇔
Et 65
36111MBDcos =−= ;
2524
25491ABCcos 2 =−=
⇒ 25
114307
2524
611
65
257DBCsin −=⋅−⋅=
75
eK)an 5
1114442CBinDs.DB.CB21SDBC −==
42AMACABBC.
21MBCsin.AM.AC
21SACM =⋅⋅==
84BC.AC21SABC ==
⇒ 5
11144425
111444284SMCD =−+−= .
60> KNnaRklaépÞ ΔXYZ
eyIgman³ BC.BA.sin21AC.AB.sin
21S 21ABC α=α= 1CB.CA.sin
21
3 =α=
AZ.AX.sin21S 1AXZ α=
92S
92
3AZ2
3ABsin
21
ABC1 ==⋅⋅α=
92SSS CZYBXYAXZ ===⇒
( )31
9231SSSSS CZYBXYAXZABCXYZ =
×−=++−= .
61> eRbobeFobRklaépÞqekaNTaMgBIr
76
tamrUbTI(1) 642ABC SSSS ===
Asin)acbcaba(21Asin)ba)(ca(
21SS 2
ABC1 +++=++=+
Asin)cba(a21SAsin.bc
21Asin)cba(a
21S ABC1 ++=−+++=⇒
dUcKñaEdr ⇒ Bsin)cba(b21S;Csin)cba(c
21S 53 ++=++=
RklaépÞrUbTI(1): ABC654321)1( SSSSSSSS ++++++= enaH ABC)1( S4Csin)cba(c
21Bsin)cba(b
21Asin)cba(a
21S +++++++++=
ABCS4)CsincBsinbAsina)(cba(21
+++++= (a)
tamrUbTI (2): Csinc21'S;Asina
21'S;Bsinb
21'S 2
62
42
2 ===
müa:geTot ABC222
ABC1 S2Asinb21Asinc
21Asin)cb(
21S'S ++=+=+
ABC22
1 SAsinc21Asinb
21'S ++=
eFVIdUcKñaEdr eK)an ABC22
3 SCsinb21Csina
21'S ++=
ABC22
5 SBsinc21Bsina
21'S ++=
77
RklaépÞrUbTI(2): ABC654321)2( S'S'S'S'S'S'SS ++++++=
enaH Bsinc21Bsina
21Csinc
21Bsinb
21Asina
21S 22222
)2( ++++=
ABC2222 S4Asinc
21Asinb
21Csina
21Csinb
21
+++++
( ) ABC222 S4CsinBsinAsin)cba(
21
+++++= (b)
tamRTwsþIbT sin kñúg ΔABC eyIg)an³
CsincBsinbAsinacba
CsinBsinAsincba
Csinc
Bsinb
Asina 222
++++
=++
++===
(c) )CsinBsinA)(sincba()CsincBsinbAsina)(cba( 222 ++++=++++⇒
tam (a); (b); (c) ⇒ )2()1( SS = . 62> bgðajfa J Cap©iténrgVg; carwkkñúg ΔCEF
eyIgman [EF] CaemdüaT½rén [OA] [EF]⊥[OA] Rtg;cMnuckNþal [OA]
rW [OA] CakaMrgVg;Ekg [EF]
⇒ [OA] Ca emdüaT½rén[EF]
enaH FCAACEFAEA =⇔=))
eK)an [CJ) CaknøHbnÞat;BuH ∠ ECF (1) eday [AO] ⊥ [EF] Rtg;cMnuckNþal; OE = OF
⇒ OEAF CactuekaNesμ I vi)ak EA = OE = AF = OF
müa:geTot D kNþal BA)⇒ [OD] ⊥ [AB]
Et [AB] ⊥ [AC] ⇒ [OD] // [AC]
78
Et [DA] // [OJ] enaH ODAJ CaRbelLÚRkam ⇒ OD=OE=AE=AJ
eK)an AEJ CaRtIekaNsm)at vi)ak ECJCEJEJAJEA +==
Et )FEAECAFAEA(JEFECJJEFFEAJEA =⇒=+=+=))
⇒ naMeGay [EJ) CaknøHbnÞat;BuH ∠FEC (2) CEJJEF =
tam (1) & (2) knøHbnÞat;TaMgBIrkat;KñaRtg; J
dUcenH J Cap©iténrgVg; carwkkñúg ΔCEF . 63> a. Rsayfa DH=DK
eyIgman AK=AB; CB=CH
⇒ ΔAKB & ΔCBH CaRtIekaNsm)at eday HBCKBA =
eK)an HCBBAK =
HCBBCDHCD
BAKBADKAD
+=
+=
Et HCDKADBCDBAD =⇒=
ΔADK & ΔDCH man AK=AB=DC CH=BC=AD HCDKAD =
⇒ ΔADK ≅ ΔDCH eK)anvi)ak DK=DH
79
b. Rsayfa ΔDKH ∼ ΔABK
tag [DK]∩[AB]={E}
eday (AH) // (DC) ⇒ DHEHDC =
Et ( ΔADK≅ΔDCH ) EKAHDC =
eK)an DHEEKA =
ΔAKE & ΔEDH man ⎪⎩
⎪⎨⎧
=
=
HEDKEA
DHEEKA⇒ HDEKAE =
Et ΔAKB & ΔKDH CaRtIekaNsm)at ⇒ ΔDKH ∼ ΔABK .
64> Rsayfa AC=ST ehIybnøayén (AC) Ekgnwg (ST)
eyIgman )SCBTCD(º360SCTBCD +−=+
º180º180º360 =−= Et º180CDABCD =+
¬mMuBIrCab;RCugmYyénRbelLÚRkam¦ ⇒ SCTCDA =
ΔADC & ΔCTS man:
CTDC;SCTCDA;CSBCAD ====
⇒ ΔADC ≅ ΔCTS vi)ak AC = ST
tag H CacMnucRbsBVrvag (AC) & (ST)
eyIgman º90º90º180SCBº180SCHBCA =−=−=+
Et ⇒ HSCCADBCA == º90SCHHSC =+
80
naMeGay º90SHC =
dUcenH bnøay (AC) Ekgnwg (ST) .
65 . RsaybBa¢ak;fa BC kat;tamcMnucnwgmYy tag H CacMeNalEkgén O elI (xy)
E CacMnucRbsBVrvag (OH) nig (BC)
F CacMnucRcsBVrvag (OA) nig (BC)
ΔOEF nig ΔHOA CaRtIekaNEkgman ∠O CamMurYm enaH ΔOEF ∼ ΔHOA
enaH OA.OFOH.OEOHOF
OAOE
=⇔= ΔOCF nig ΔOCA CaRtIekaNEkgman ∠COA CamMurYm enaH ΔOCF ∼ ΔOCA vi)ak 22 rOCOF.OA
OCOF
OAOC
==⇔=
efr ==⇒ 2rOH.OE
eday [OH]efr ⇒ OE efr ehIy EenAelI [OH]
dUcenH E CacMnucnwg ⇒ (BC) kat;tamcMnucnwgmYyCanic©.
66 . Rsayfa 1FACF
GCBG
EBAE
=⋅⋅ tag O CacMnucRbsBVrvag AG ; BF ; CE. tam A KUsbnÞat;Rsb(BF)kat;bnøay(CE)Rtg; I tam C KUsbnÞat;Rsb(BF)kat;bnøay(AG)Rtg; J
81
eyIgman ( ) ( )OICO
FACFIA//OF =⇒
Δ OAI ~ Δ JCO ( (JC) // (IA) )
AICJ
OICO
=⇒
enaH IAJC
FACF
= Δ EIA ~ Δ EOB ((OB) // (IA))
OBIA
EBAE
=⇒
Δ BOG ~ Δ GJC ((BO) // (JC))
JCOB
GCBG
=⇒
eyIg)an : 1AICJ.
JCOB.
OBIA
FACF.
GCBG.
EBAE
==
dUcenH 1FACF.
GCBG.
EBAE
= .
67 . Rsayfa A1; B1; C1 rt;Rtg;Kña eyIgman : ( ) ( ) ( ) ( )1111 BBPB;PAAA ⊥⊥ ⇒ CactuekaNcarwkkñúgrgVg;. PBBA 11
enaH º180BPACBA 11 =+
Et º180CPACBA =+
eyIg)an CPABPA 11 =
( )1CPBAPA 11 =⇒ müa:geTot ( ) ( ) ( ) ( )PCAC;PAAA 1111 ⊥⊥ CactuekaNcarwkkñúgrgVg;. APCA 11⇒
82
enaH ( )2APAACA 111 = ehIy PCAPBA;PBAPBA 1111 ==
enaH PCAPBA 11 =⇒ ( )3CPBCCB 111 =
tam ( ) ( ) ( ) CCBACA3;2;1 1111 =⇒
dUcenH rt;Rtg;Kña. 111 B;C;A
68> RsaybBa¢ak;fa PNMMNQ = yk I CacMnucRbsBVrvag (AC) & (MN)
enaH I CacMnuckNþal [AC]
tam I KUsbnÞat;Ekg [MN] ehIykat; (QN) Rtg; K
eK)an KM = KN ⇒ ΔKMN CaRtIekaNsm)at vi)ak (1) NMKMNQ =
eday M & N enAkNþal [AB] & [DC]
⇒ (MN)//(BC) ⇔ ICQI
MPQM
=
müa:geTot (IK)//(DC) ⇒ KNQK
ICQI
=
⇒ KNQK
MPQM
= ; vi)ak (MN)//(PN)
eyIgTaj)an (2) PNMNMK =
tam (1) & (2) dUcenH . PNMMNQ =
69> Rsayfa PB = QC
83
tag sDQ&rAP,yQC,xPB ==== eyIgman ysxrCDAB +=+⇒= (1)
sDQQV,rPAPU ==== yCQQU,xPBPV ==== enaH rxPUPVUV −=−= syDQQUUV −=−= eK)an syrx −=− (2)
tam (1) & (2) ⇒ x = y
dUcenH PB = QC . 70> Rsayfa (XY)//(BC)
tag U & V CacMnucenAelI AB & AC Edl BA92BU = , CA
92CV =
eK)an CABA
CVBU
= ⇒ (UV)//(BC) (1)
müa:geTot PRPX
31
23.
92
BRBA.
BABU
BRBU
==== ⇒ (UX)//(BC) (2) dUcKñaEdr eK)an (YV)//(BC) (3)
tam (1), (2) & (3) ⇒ X, Y enAelI (UV)
dUcenH (XY)//(BC) . 71 . RsaybBa¢ak;fa : IE CD⊥
84
tag O CacMnuckNþalénRCug BC. tamRbBn½§kUGredaeneyIg)an : O(0,0) ; A(0,a) ;
B(-c,0) ; C(c,0) ; ⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛−
2a,
6cE;
2a,
2cD
( )a,cAB −− . eday Δ ABC CaRtIekaNsm)atkMBUl C
enaHtagp©itrgVg;carwkeRkAKW I (0,y)
eK)an : ⎟⎠⎞
⎜⎝⎛ −−= y
2a,
2cID
ID ⊥ AB naMeGay 0AB.ID = , ⇒ ( ) 0y2aac
2c
=⎟⎠⎞
⎜⎝⎛ −−−−
a2cay
22 −=⇒ . naMeGay ⎟⎟
⎠
⎞⎜⎜⎝
⎛ −a2ca,0I
22
eK)an : ⎟⎟⎠
⎞⎜⎜⎝
⎛a2
c,6cIE
2
nig ⎟⎠⎞
⎜⎝⎛ −
2a,
2c3DC
naMeGay : 04c
4c
2a.
a2c
2c3.
6cDC.IE
222
=−=−= . dUcenH IE ⊥ DC .
72 . RsaybBa¢ak;fa ebI MNPQ CakaerenaH ABCDk¾CakaerEdr tag : aAB,xDPAQBM ====
. α=−== DQP;xaAM;uMN
tam Δ AMQ eyIg)an : ( ) ( ) Acosxax2xxau 222 −−+−= (1)
85
kalNa MNPQ CakaerenaH : nig α−= º90MQA uNMPNPQMQ ==== eyIg)an : MQAcos.QM.AQ2QMAQAM 222 −+=
(2) ( ) α−+=−⇔ sinux2xuxa 222
tamRTwsþIbTsuInuskñúg Δ QDP eyIg)an : DsinxsinuDsin
usin
x=α⇒=
α
ykCMnYskñúg (2) eyIg)an : ( ) Dsinx2xuxa 2222 −+=− (3)
tam (1) nig (2) naMeGay : ( ) 0Dsin1xa
xAcos ≥−−
= . ctuekaN ABCD CactuekaNe)a:gkalNa : º90A0 ≤< . RsaydUcKñaEdreyIg)an : º90D0;º90C0;º90B0 ≤<≤<≤< . º360DCBA ≤+++⇒ naMeGayeyIg)an : A = B = C = D = 90º . dUcenHctuekaN ABCD Cakaer. 73 . RsaybBa¢ak;faebI PM CD⊥ enaH QM Edr AD⊥
tag bMB,aMArr . ==
eyIgman Δ MAB ∼ Δ MDC naMeGay kMBMC,kMAMDk
MBMC
MAMD
==⇒==
dUcenH bbkaMD,a
akbCM
rr −==
cMeBaHlkçx½NÐ a = MA nig b = MB. Eteday CDMDCM =+ naMeGay ( )bacb
abkCD 22
rr−= .
tamsmμtikmμ, PM ⊥ CD naMeGay 0MP.CD = ,ehIy MP2MBMA =+
86
naMeGay ( )( ) 0baabba 22 =−+rrrr eyIg)an ( ) 0abba 22 =−
rr . + ebI ,ehIy a = b enaH ABCD CactuekaNBñaysm)at 0ab 22 =−
ehIyCactuekaNEkg. + ebI enaH AC ⊥ CD naMeGayeyIg)an : 0ba =
rr
( )
( )ODOA21OQOMQM
OMODOCOBOA21
+=−=
=+++
eyIg)an : ( )( ) ( ) 0OAOD
21OAODODOA
21AD.QM 22 =−=−+= .
dUcenH QM ⊥ AD . 74> KNna R CaGnuKmn_én r1 & r2
snμt; r2 > r1
tag CacMenalEkgén S1S′ 1 elI [AB]
⇒ AB S1S′ 1 CactuekaNEkg )º90ADSDAS( 11 ==′
tag CacMenalEkgén S2S′ 2elI [AB]
⇒ S2S′ 2BC CactuekaNEkg
)º90SCBCBS( 22 ==′
tag H CacMenalEkgén S1 elI [ 22SS ′ ]
⇒ CactuekaNEkg 22SS ′ 11SS′ )º90SSSSSS( 221211 =′′=′′
87
tag P; Q; R CacMnub:HerogKñaénrgVg;TaMgbI eyIgman 111 rRSDSAASSASS −=−=′−=′ 2222 rRCSSBBSSBSS −=−=′−=′ ⇒ )rr(R2SBSAABSSSS 21212121 +−=′−′−=+′
kñúg Δ man³ 11SSS ′ 211
21
21 SSSSSS ′+′=
12
12
12
12111 Rr2)rR()rR(SSSSSS =−−+=′−=′
kñúg Δ man³ 22SSS ′ 22
22
222 Rr2)rR()rR(SS =−−+=′ müa:geTot 2112222 HSSSHSSHSS +′=+′=′ ⇔ 2Rr2 = 1Rr2 +HS2
kñúg Δ man³ HSS 21
HS2= 221
221
221
221
21
221 )]rr(R2[)rr(SSSSHSSS +−−+=′′−=−
221 R4)rr(R4 −+= = )Rrr(R2 21 −+
enaH )Rrr(R2Rr2Rr2 2112 −++= ⇔ Rrrrr 2112 −+=−
⇒ 21rr2R = . 75> KNna CD
tag E CacMnucRbsBVénbnøay [BC] & [DA] kñúg ΔECD & ΔABD man DCEDBA;CDEDAB == ⇒ ΔECD ≈ ΔABD (1) vi)ak ⇒ BED CaRtIekaNsm)at BDEBED=
eK)an BE = BD =10 cm
88
tam (1) ⇒ BD
)BEBC(ABBD
ECABDCECBD
DCAB +
=⋅
=⇒=
eK)an 5
6410
)610(8DC =+⋅
= .
76> bgðajfa AB + DB = BC
man AB = AC ⇒ ABC CaRtIekaNsm)at ⇒ º40
2º100º180ABCBCA =
−==
man º202
CBACBDDBA === kñúg ΔADB man³
º100sinº20sinBDAD
º100sinBD
º20sinAD
º60sinAB
BDº100sin
ADº20sin
ABº60sin
++
=++⇔==
º40cos2
BDADº40cosº60sin2
)BDAD(º60sinAB +=
+=⇔
kñúg ΔABC man³ º100sin
BCº40sinABº40sin
ABº100sin
BC=⇒=
º80sin
)DBAD(º100sinº40cosº40sin2
)DBAD(º100sinBCº40cos2
DBADº100sin
BCº40sin +=
+=⇔
+=⇒
eK)an BC = AD + DB (sin100º=sin80º) .
77> RsaybBa¢ak;faebI D CacMnuckNþal [AB] enaH DM = DL tag E CacMnuckNþal [AP]; F kNþal [BP] eday D kNþal [AB]
eK)an [DE] & [DF] Ca)atmFüm ΔABP
89
⇒ (DE) // (BP); BP21DE ⋅=
(DF) // (AP); AP21DF ⋅=
naMeGay DEPF CaRbelLÚRkam; vi)ak PEDPFD;EPDF;FPDE ===
Δ⊥PBL man F kNþal [BP] ⇒ PF=LF=DE; LBP2LFP =
Δ⊥PAM man E kNþal [AP] ⇒ ME=PE=DF; MAP2MEP =
müa:geTot LFPMEPMAPLBP =⇒=
⇒ MEPPEDMED
LFPPFDLFD
+=
+= MEDLFD =
kñúg ΔDFL & ΔDEM man LF=DE; ME=DF; MEDLFD =
naMeGay ΔDFL ≅ ΔDEM; vi)ak DMDL = .
78> Rsayfa BM=AC
eday CM=CB ⇒ ΔCMN CaRtIekaNsm)at º40
2BCAº180BMC =
−=
kñúg ΔABC: º50sin2
ACBCAC
º50sinBC
º30sin=⇒=
kñúg ΔMCN: º100sin
º40sin.BMBCBM
º100sinBC
º40sin=⇒=
eK)an º50cosº50sin2
º40sin.BMº100sin
º40sin.BMº50sin2
AC==
⇒º50cos
º40sin.BMAC = Et 50º+40º=90º ⇒sin40º=cos50º
dUcenH AC = BM .
90
79> KNna CMA
tag HC CakMBs; ΔABC; (MB)∩(CH)={N}
eday [CH] CakMBs;énRtIekaNsm)at ⇒ [CH] CaknøHbnÞat;BuHmuM & CaemdüaT½r eK)an NA=NB ⇒ º30BAN =
º40NBABANAMN
º20º10º30MABBANNAM
=+=
=−=−=
ΔAMN & ΔNCB man º40BCNAMN == ⎟⎟⎠
⎞⎜⎜⎝
⎛== º40
2BCABCN
NBAN
CBNº20MAN=
==
⇒ ΔAMN ≅ ΔNCB
vi)ak MN=NC ⇒ ΔNMC CaRtIekaNsm)at Δ⊥NHB man º60BNHº30NBH =⇒=
enaH º30CMNº60CMN2 =⇔=
dUcenH . º70º30º40CMNNMACMA =+=+=
80> Rsayfa p2 + q2 = c2 + d2 + 2ab
tag E; F CacMenalEkgén A,B elI [DC]
⇒AEFB CactuekaNEkg; vi)ak AB=EF=a
eyIgman 2222 pECAEAC =+=
2222 qDFBFBD =+=
91
22 qp + 2222222222 DFFCBCECDEADDFBFECAE +−++−=+++= 222222 FCDEDFECdc −−+++=
222222 FCDE)FCDC(ECdc −−−+++= 222222 FCDE)FCb()FCa(dc −−−++++=
22222222 FCDEbFC2aFC2FCFCbadc −−−++++++= )ab(FC2DEFCab2)ba(dc 22222 −−−++−++=
ab2)FCDE(FC2DEFC)FCDE(dc 22222 ++−−++++= ab2FC2DE.FC2DEFCFCFC.DE2DEdc 2222222 +−−−+++++=
eK)an ab2dcqp 2222 ++=+ .
81> bgðajfa PB=2PD
eyIgman nig BCABPA =
(s>k) BCAPDA =
⇒ PDABPA =
kñúg ΔAPB & ΔAPD man³ ∠A CamuMrYm ⇒ ΔAPB ∼ ΔAPD PDABPA =
vi)ak 22 AD4AB.ADAPAPAD
PBPD
ABAP
==⇒==
⇒ AP=2AD naMeGay 21
AD2AD
PBPD
== ⇔ PB=2PD .
82> Rsayfa PA=PB+PC
92
tag M;N CacMnucelIRCug [PA]
Edl BPBN;CPCM == (1)
eyIgman º60ABCAPC ==
(2) º60BPABCA ==
tam (1) & (2)⇒ ΔBPN & ΔCMP
CaRtIekaNsm½gS müa:geTot BCMº60BCMCBAACM −=−=
BCMº60BCMMCPBCP −=−= ⇒ BCPACM =
Et ACMBAPBAPBCP =⇒=
eFVIdUcKñaEdr ⇒ ABNMAC =
ΔAMC & ΔANB man ABNMAC;ABCA;BANACM ===
⇒ ΔAMC ≅ ΔANB vi)ak CM=AN (a)
Et AP=AN+NP; CM=PC; NP=PB (b)
(a) & (b)⇒ AP=PB+PC .
83> Rsayfa PCD CaRtIekaNsm½gS tag H; E CacMenalEkgén P elI (AB) & (AD) ⇒ PE=AH; AE=PH Et º15ABPBAP ==
⇒ PAB CaRtIekaNsm)at
93
vi)ak 2
ABPEAH == ΔAPD & ΔPBC man AP=PB; AD=BC
º75º15º90PAD =−= º75º15º90PBC =−= ⇒ΔAPD ≅ ΔPBC
vi)ak PD=PC (1)
müa:geTot ABº15cos2
º15sinAHº.15tanPHPHAHº15cot ⋅==⇒=
ABº15cos2
º15sinABAEADED ⋅−=−=
⎟⎠⎞
⎜⎝⎛ −=
º15cos2º15sin1AB
⎟⎠
⎞⎜⎝
⎛−=
º15sinº15cos2º15sin31AB
2
AB23)º15sin21(AB 2 ⋅=−=
ΔPEB man 222
222 AB4
AB4
AB3PEEDPD =+=+= ⇒ PD=AB (2) tam (1) & (2) ⇒ PCD CaRtIekaNsm½gS.
84> KNna FB
ΔAFC & ΔBCE man³ AF=BC=2 AC=BE=4 ⇒ ΔAFC ≅ ΔBCE FC=CE vi)ak CBEBAF =
94
ΔAFB & ΔBED CaRtIekaNsm)atman CBEBAF =
enaH ΔAFB ∼ ΔBED vi)ak DEBF
BDAB
=
⇒ 14
22BD
DE.ABBF =×
== ; dUcenH BF=1 .
85> KNna |ST|
yk R > r
tag H CacMenalEkgBI O→(O´T)
eK)an OSTH CactuekaNEkg vi)ak ST = OH
Δ⊥OHO´: 22 H'O'OOOH −=
r'R2)rR()rR(OH 22 =−−+= dUcenH Rr2ST = .
86> KNna |AED|
eyIgman α⋅⋅= sinEBEA21AEB
β⋅⋅= sinEDEA21AED
eday β=α⇒=β+α sinsinº180
EBED3
EBED.AEB
AEDEDEB
AEDAEB
⋅==⇒=
dUcKñaEdr ⇒EBED
BECCED
EDEB
CEDBEC
=⇒=
95
⇒ 3AED10AED
EBED
AED210
⋅=⇒=
dUcenH 15AED = .
87> rkRklaépÞ ΔABC
tag x = BC
eyIgman 21
SSS
21S
ABC
APBABCAPB =⇔=
eday ΔAPQ ∼ ΔABC
⇒21
xPQ
2
2
= 2
x2PQ =⇔
eday ΔAPQ; ΔSCR; ΔTUB manRCugTaMgbIRsbKñaerogKña ehIy TUBSCRAPD SSS == eK)an ΔAPQ ∼ ΔSCR ∼ ΔTUB
enaH PQ = CR
eday RB+CR=BC ⇒RB=BC-CR= x2
222
x2x ⋅⎟⎟⎠
⎞⎜⎜⎝
⎛ −=−
eday (RS)//(AB); (PQ)//(CB); (TU)//(AC)
⇒ PXUC; YQBR CaRbelLÚRkam Et CU = RB enaH x
222RBYQPX ⋅
−===
x22
23x2x22
x2PX2PQXY ⎟⎟⎠
⎞⎜⎜⎝
⎛−=+−=−=
96
221217
x
x22
23
BCXY
SS
2
22
2
2
ABC
XYZ −=
⎟⎠
⎞⎜⎝
⎛−
==
eK)an 22434SABC += . 88 . KNnaRklaépÞénctuekaN PQRS
eyIgman : ABCACDABCD SSS +=
Et CMBABCADKACD S2S;S2S ==
enaH ( )CMBADKABCD SS2S +=
2SSS ABCD
CMBADK =+⇒ müa:geTot : ABCDPQRSSRCKAMQPCMBADK SSSSSS =++++
5993885132
3000S
SS2
SS
SSSS2
S
PQRS
SRCKAMQPABCD
PQRS
ABCDPQRSSRCKAMQPABCD
=−−=⇒
−−=⇔
=+++⇔
dUcenH 599SPQRS = .
89 . KNna BC
eyIgman : º60DCADBA ==
enaHctuekaN ABCD CactuekaNcarwkkñúgrgVg;. tag (O;R) CargVg;Edl ABCD carwkkñúg
97
tag I elIrgVg;EdlCacMnucRbsBVrvag (BE) nig rgVg; º30IBA =⇒
yk CacMnucRbsBVrvag (CF) nig rgVg; 1I
enaH (1) º30ICAº30DCI 11 =⇒=
Et (2) º30IBAICA ==
tam (1) nig (2) naMeGay enAelI I . 1I
tamRTwsþIbTsuInus : 1
23.2
3º60sin2
DR ==Α
=
müa:geTot : CDA21º30CDA
21IBAFDAADIFDI +=+=+=
Et CDA21º30DCFCDFDFI +=+=
enaH IFIDDFIFDI =⇒=
Δ OID man º60DBI2DOI;IDOI ===
enaH Δ IOD CaRtIekaNsm½gS ; ID = OD = 1 eK)an IF = 1. RsaydUcKñaEdreyIg)an IE = 1. tamRTwsþIbTkUssIunuskñúg Δ IFE
1BCº60FIE21
IF.IE2EFIEIFFIEcos
222
=⇒=⇒=−+
= . 90. Rsayfa SCPRQ = SABR
tag 22 qpABqBC;pAC +=⇒==
98
eyIgman : KLCP
BKBC
=
qp
p.qBK
KL.BCCP+
==⇒
eFVIdUcKñaEdrqp
p.qCQ+
=⇒
( )
2q
ABq.q.AB
21
CBAcosq.AB21CBAº90sinq.AB
21NBAsinBN.AB
21S
2
ANB
==
=+==
eFVIdUcKñaEdr2pS
2
ABL =⇒
( )qp
p21
qppqp.p
21PCAC.AL
21AP.AL
21S
3
APL +=⎟
⎠⎞
⎜⎝⎛
+−=−==
( )qp
q21
qppqq.q
21QCBC.BN
21BQ.BN
21S
3
QBN +=⎟
⎠⎞
⎜⎝⎛
+−=−==
qp
qp21
qppq.p
21CQ.AC
21S
2
ACQ +=
+==
qp
pq21CB.CP
21S
2
PBC +==
tag RQBAPR SSS +=
( )SSSSS21S DBNALPABNALBRAB −−−+=
⎟⎠
⎞⎜⎝
⎛−+
+−
+=⎟
⎠
⎞⎜⎝
⎛−
++
−+
= S2
pq2
qp2
qp21S
qpqp
21
2qp
21 22223322
⎟⎠⎞
⎜⎝⎛ −= S
2pq
21
( ) ⎟⎠⎞
⎜⎝⎛ −=⎟
⎠
⎞⎜⎝
⎛−
++
=−+= S2
pq21S
qpqppq
21
21SSS
21S
22
CBRCAQCPRQ
99
dUcenH . RABCPRQ SS =
91 . Rsayfa
AFAD
AEAB
AGAC
+= tag I ; J CacMnucRbsBVrvag (l) nwg (BC);(CD)
Δ AEG ~ Δ GCJ ( )JCGGAE;EGACGJ ==
GEJE
AGAC
=⇒
GIC~AGF ΔΔFGIF
AGAC
=⇒
eK)an : ( )1EF
IFIEFGEGIFIE
FGIF
EGJE
AGAC +
=++
===
EFJE
AFADFJD~AEF
FEIF
BEABEIB~AEF
=⇒ΔΔ
=⇒ΔΔ
eK)an : ( )2EF
JEIFAFAD
AEAB +
=+
tam (1)nig (2) AFAD
AEAB
AGAC
+=⇒ . 92> RsaybBa¢ak;fa
213 R1
R1
R1
+=
+tag E, H, F CacMnucb:HnwgbnÞat; rbs;rgVg;TaMg 3. +I, J CacMenalEkgén O3 elI (O1E) & (O2F).
100
+K CadcMenalEkgénO2elI (O1E)
+sn μt 321 RRR >>
eK)an 211 RRKO −=
212
212
212
12212 RR2)RR()RR(KOOOKO =−−+=−=
kñúg Δ⊥IO1O3: 312
312
3112313 RR2)RR()RR(IOOOIO =−−+=−=
kñúg Δ⊥JO3O2: 32
232
232
22
2323 RR2)RR()RR(JOOOJO =−−+=−=
eday KOEF;HFJO;EHIO 233 === enaH 233 KOJOIO =+
⇔ 213231 RRRRRR =+ ⇒ 213 R
1R1
R1
+= .
93> KNnatMélEdlGacmanrbs; AE
]bmafa A, B, C, D, E CacMnuczitkñúgtMruyGrtUNma:l; ehIymankUGredaen )e4,e4(E);d4,d4(D);c4,c4(C);b4,b4(B);a4,a4(A 2121212121
⇒ )d2c2,d2c2(H);c2b2,c2b2(G);b2a2,b2a2(F 22112211221 ++++++ )e2d2,e2d2(I& 2211 ++ ⇒ kUGredaenén X & Y KW³ )edcb,edcb(Y);dcba,dcba(X 2222111122221111 ++++++++++++
eK)an 222
211 )ea()ea(XY −+−= ; 2
222
11 )ea()ea(4AE −+−= ⇒ XY4AE =
eday ehIy 5AE1 ≤≤ ∗Ν∈XY
enaH XY = 1 ⇒ AE = 4 .
101
94> KNna BCA tag H CacMenalEkgén C elI (AP)
tamRTwsþIbT sin kñúg ΔABP:
º15sin3
BCº45sin
AP=
⇒ º15sin3
BCº45sinAP =
ΔCHP CaRtIekaNknøHsm½gS
⇒ 3
BC23BC2
2PCPH ===
3
3BC2
3PCHC == (1)
müa:geTot HPAPAH −=
3BC1
º15sinº45sin
3BC
º15sin3BCº45sin
⎟⎠⎞
⎜⎝⎛ −=−=
3
3BC3
BCº15sin
º15sinº45sin=⎟⎠
⎞⎜⎝⎛ −
= (2)
tam (1) & (2) naMeGay AH = HC ⇒ º45ACH =
dUcenH . º75ACHHCPBCA =+=
95> KNna λ
eyIgman ACAM
CECN
= eday CE = AC ⇒ CN = AM ⇒ EN = CM
102
ΔEND & ΔCMB man CMEN,BCED,CE ===
⇒ RtIekaNTaMgBIrb:unKña ⇒ NDECBM =
& CMBDNE =
tag β=α= CMB,CBM
enaH α−=β−= º90BNC,º180CND
º150º30º180BCMº180 =−=−=β+α
müa:geTot
yk I zitenAelIrgVg;pí©t C kaM BC ⇒ º120)º90()º180(BNCCNDBND =α−+β−=+=
º60DIB =
eday ⇒ C, N, D, B sßitenAelIrgVg;EtmYy º180DNBDIB =+
⇒ CN = BC = R kñúg ΔOCE : =−+= COEcos.OC.OE.2OCOEEC 222
R3ECR3OE3º120cosOE2OE2 2222 =⇒==−=
dUcenH 33
R3R
CECN
===λ .
96> KNnaRbEvg AB
tag G CaeCIgkMBs;KUsecjBI D eTAelI JH
C CacMnucRbsBVrvagbnÞat;KUsecjBI B ehIyRsb ID nwgbnÞat;KUsecjBI A ehIyRsb DG
E CacMnucRbsBVrvagbnøay ID & AC
I CaeCIgkMBs;KUsecjBI B elI IDman G
103
CaeCIgkMBs;én ΔDH
215GHDHDG&
21
2KHGH 22 =−===⇒
kñúg ΔJAH & ΔAHG man ∠H CamMurYm nig 21
AHGH
JHAH
==
⇒ ΔJAH ∼ ΔAHG Et JAH CaRtIekaNsm)at ⇒ ΔAHG CaRtIekaNsm)at ⇒ GF=DE=1/4
müa:geTot ΔILB ≅ ΔAHF ⇒ IL=FH=1/4, IB=AF=415
FHAH 22 =−
2LDGFILLDDEILLDIEBC ==+−=+−==
15215
415
.2DGAF2AFDGIBAFEFCEAC =+=+=++=++=
19CABCAB 22 =+=⇒ . 97> Rsayfa MCBDCA = edAcMnuc N Edl (BM)//(CN) & CN = BM
⇒ BNCM & ANCD CaRbelLÚRkam ⇒ NCBCBMMDC ==
ΔABN & ΔMDC man MCBN,DMAB == & DCAN = ⇒ ΔABN ≅ ΔMDC
vi)ak ⇒ MDCNAB = MDCNCB =
⇒ A, B, N, C enAelIrgVg;EtmYy ⇒ NACNBC =
Et ⇒ DCANAC&MCBNBC == MCBDCA = .
104
98> KNnakaMénrgVg;tUc tag x CargVas;kaMrgVg;tUc edayrgVg;FMb:HknøHrgVg;Rtg;p©it C ⇒ AC = 1 - x1DCACAD −=−= - x1FBAFAB +=+= - 222 DCCBDB −= x44x)x2(x)BGCG( 2222 −=−−=−−=
eday ΔADB CaRtIekaNEkg ⇒ ⇒ 222 DBADAB += x44)x1()x1( 22 −+−=+
⇒ 2/1x = . 99> Rsayfa AE = EB + BC edA B' enA (AB) Edl AB' = BC
kñúg ΔADB' & ΔDBC man AB' = BC, AD = DC & CA =
⇒ ΔADB' ≅ ΔDBC vi)ak DB = DB'
ΔDB'E & ΔDEB CaRtIekaNEkg man DE CaRCugrYm nig DB = DB' ⇒ ΔDB'E ≅ ΔDEB vi)ak EB = EB'
müa:geTot AE = AB' + EB' = BC + EB
dUcenH AE = EB + BC .
105
100 . rktMélGb,brmaénplbUk : S = zc
yb
xa
++
eyIgman : > . enAelIRCug BC edAcMnuc CPB ACB
E Edl : ACPBEPACBCPE =⇒=
ehIy : CAPEBP =
∼ (m-m) PBEΔ⇒ PACΔ
eyIgman : nig CEPABP = BAPECP =
∼ PBAΔ⇒ PECΔ
( ) ( )2PKCE
PMAB,1
PKBE
PLAC
==⇒
yk (1) + (2) eyIg)an :
xa
zc
yb
PKBC
PKCEBE
PMAB
PLAC
=+⇒=+
=+
eyIg)an : xa2
zc
yb
xaS =++= .
dUcenH S Gb,brma kalNa x Gtibrma. eyIg)an : RPOPK =≤
PK Gtibrmaesμ I R kalNa K ≡ O ⇒ min S = 4.
101. k ) bgðajfa : AD + BC = CD
enAelIRCug DC eKedAcMnuc F Edl DF = DA
ebI F ≡ E ⇒ CD = CE + ED = CB + DA
eyIg)an : AD + BC = CD
ebI F ≠ E : enaH F sßitenAcenøaH D nig E
106
eyIg)an : ( )
2BFAD180
2CBAAFD180EBAEFA ºº +−=+−=−
B180180AFD2
B180FAD2180CDAºº
ºº
+−=⇒
−=−=⇒
º180EBAEFA2BFADAFD =+⇒==⇒
⇒ctuekaN ABEF carwkkñúg 2AFBC,
2ABAEBFA ===⇒
Casm)atkMBUl C ⇒ CB = CF BCFΔ⇒
eyIg)an : CD = CF + FD = BC +AD
x ) KNna BCE
ADE
SS
edayBIcMnuc E esμ IcMgayeTARCug AD , AB , BC
1k1BCCD
SS
BCBCCD
BCAD
SS
BCE
ADE
BCE
ADE −=−=⇒−
==⇒
dUcenH 1kSS
BCE
ADE −= .
102> Rsayfa ΔABC CaRtIekaNsm½gS
kñúg ΔABC: cba ch21bh
21ah
21S ===
enaH S2c
h1;
S2b
h1;
S2a
h1
cba===
tam Heron: )cp)(bp)(ap(pS −−−=
107
enaH S
)cp)(bp()ap(p
1 −−=
−;
S)cp)(ap(
)bp(p1 −−
=−
S
)ap)(bp()cp(p
1 −−=
−
naMeGayS
)ap)(cp()cp)(bp()bp)(ap(S2
cba −−+−−+−−=
++
)ap)(cp()cp)(bp()bp)(ap()cba(21
−−+−−+−−=++⇔ (1)
tam Cauchy: )cp)(p(2cpbp −−≥−+−
)cp)(p(2bcp2 −−≥−⇔ )cp)(bp(
2a
−−≥
dUcKñaEdr )cp)(ap(2b
−−≥
)bp)(ap(2c
−−≥
eK)an )ap)(cp()cp)(bp()bp)(ap()cba(21
−−+−−+−−≥++ (2)
tam (1); (2) eK)an a = b = c dUcenH ΔABC CaRtIekaNsm½gS.
103 . Rsayfa D enAcenøaH M & H
eyIgman : 222 AHABBH −= 222 AHACCH −=
108
Et ( )CBACAB >< ( )1BM
2BCBHCHBH =<⇔<⇒
müa:geTot ABCº90HAB −=
HCAº90CAH −=
enaH ( )CBCAHHAB >< DAB
2CABHAB =<⇒
eK)an BH < BD (2)
[AD) CaknøHbnÞat;BuHmuM A 1ACAB
DCBD
<=⇒
( )3BM2
BCBDDCBD =<⇔<⇒
tam (1) ; (2) nig (3) naMeGay D enAcenøaH H nig M . 104> Rsayfa
Rr
P'P=
tag O Cap©itrgVg;carwkeRkA ΔABC
H CaeCIgcMenalEkgBI O elI BC
eyIgman AcosABAE,AcosACAF == ⇒
ABAC
AEAF
=
ehIy ∠A CamMurYmrvag ΔABC & ΔAFE ⇒ ΔABC ∼ ΔAFE
vi)ak AcosBCFEAcosACAF
BCFE
=⇒==
109
dUcKñaEdreK)an CcosABDE,BcosACFD == brimaRtén ΔEDF KW CcosDEBcosACAcosBCDEFDFE'P ++=++= müa:geTot BC.AcosR
21BC.OH
21SOBC ==
dUcKñaEdreK)an AB.CcosR21S,AC.BcosR
21S OABOAC ==
eday OABOBCOACABC SSSS ++= & rP21rpSABC ==
⇒ )AB.CcosAC.BcosBC.A(cosR21rP
21
++=
⇔ Rr
P'P'RPrP =⇒= .
105> RsaybBa¢ak;fa ΔABC CaRtIekaNsm½gS eyIgman )ba(c)ac(b)cb(a)cba(2 222222333 +++++=++
333
222222222333
cbaab2
cbaabc2ac2
bcaabc2bc2
acbabc2)cba(2
+++
−++
−++
−+=++⇔
)CcosBcosA(cosabc2cba 333 ++=++⇔
eyIgnwgRsayfa 23CcosBcosAcos ≤++
23)BAcos(
2BAcos
2BAcos2
23CcosBcosAcos −+−
−+=−++
231
2BAcos2
2BAcos
2BAcos2 2 −+
+−
−+=
110
⎥⎦⎤
⎢⎣⎡ +
−+−
+−=
41
2BAcos
2BAcos
2BAcos2 2
0
2BAsin
41
2BAcos
21
2BAcos2 2
2
≤⎥⎥⎦
⎤
⎢⎢⎣
⎡ −+⎟⎠
⎞⎜⎝⎛ −
−+
−=
⇒23CcosBcosAcos ≤++
⇒ abc323.abc2cba 333 =≤++ (1)
tamvismPaB Cauchy ⇒ (2) abc3cba 333 ≥++
tam (1) & (2) eK)an abc3cba 333 =++
smPaBekIteLIgkalNa cba == dUcenH ΔABC CaRtIekaNsm½gS . 106> RsaybBa¢ak;fa ΔABC CaRtIekaNsm½gS tamsmμtikmμ eyIg)an
4R81)mmm(
22
cba =++ (1)
eyIgBinitü ³ )mmmmmm(2mmm)mmm( accbba
2c
2b
2a
2cba +++++=++
)mmm(3 2c
2b
2a ++≤
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−++⎟
⎠
⎞⎜⎝
⎛−++⎟
⎠
⎞⎜⎝
⎛−+=
4c
2b
2a
4b
2a
2c
4a
2c
2b3
222222222
)cba(49 222 ++=
(2) )CsinBsinA(sinR9 2222 ++=
tam (1) & (2) eK)an³
111
4R81)CsinBsinA(sinR9
22222 ≥++
⇔49CsinBsinAsin 222 ≥++
⇔49CcosBcosAcos22 ≥+
⇔ 041)bAcos()BAcos()BA(cos2 ≤+−+++
⇔ 0)BA(sin41)BAcos(
21)BAcos( 2
2
≤−+⎥⎦⎤
⎢⎣⎡ −++ (*)
eyIgeXIjfa (*) GacekItmancMeBaHsBaØa “=”
eK)an cba mmm ==
dUcenH ΔABC CaRtIekaNsm½gS 107> Rsayfa 0
2Bcot)ac(
2Acot)cb(
2Ccot)ba( =−+−+−
eyIgman 2Bcos
2Csin
2Ccos
2Bsin
2Csin
2Bsina
2C
2Bsin
2Csin
2Bsina
2Acos
2Csin
2Bsina
r+
=⎟⎠⎞
⎜⎝⎛ +
==
⇒ )2Ccot
2B(cotra +=
dUcKñaEdr eK)an ⎟⎠⎞
⎜⎝⎛ +=
2Ccot
2Acotrb
⎟⎠⎞
⎜⎝⎛ +=
2Acot
2Bcotrc
⎟⎠⎞
⎜⎝⎛ −=−•
2Acot
2Bcotr)ba(
112
⎟⎠⎞
⎜⎝⎛ −=−•
2Bcot
2Ccotr)cb(
⎟⎠⎞
⎜⎝⎛ −=−•
2Ccot
2Acotr)ac(
eK)an 02Bcot)ac(
2Acot)cb(
2Ccot)ba( =−+−+− .
108> kMnt;RbePTRtIekaN ABC
a. )cba)(cba(41S +−−+=
eyIgman )acb(21ap −+=−
)bac(21bp −+=−
)cba(21cp −+=−
tamsm μtikm μ eK)an³ )cba)(cba(
41)cp)(bp)(ap(p +−−+=−−−
⇔ 22 )cba()cba(161)cba)(bac)(acb)(cba(
161
+−−+=−+−+−+++
⇔ )cba)(cac()acb)(cba( −+−+=−+++ ⇔ 222 cba +=
naMeGay ΔABC CaRtIekaNEkg . 2)cba(
363
++= b. S
tamRTwsþIbT Cauchy:
113
3
3cpbpap)cp)(bp)(ap( ⎟⎠⎞
⎜⎝⎛ −+−+−
≤−−−
⇔27p)cp)(bp)(ap(
3
≤−−−
⇔27p)cp)(bp)(ap(p
4
≤−−−
⇒ 22 )cba(36
3p33
1S ++=≤
smPaBekIteLIgkalNa cpbpap −=−=− ⇒ ΔABC CaRtIekaNsm½gS . 109> Rsayfa rR'OC'OB'OA +=++ tag γ=β=α= 'OC,'OB,'OA eyIgman OACOBCOABABC SSSS ++= ⇔ )cba(
21)cba(
21
γ+β+α=++
⇒ cbacbar
++γ+β+α
= (1)
müa:geTot AcosRR
Acos =α⇒α
= ehIy CsinR2c,BsinR2b == ⇒ [ ] )CsinAcosBsinA(cosR2)CsinB(sinAcosR2)cb( 22 +=+=α+
dUcKñaEdreK)an )AsinBcosCsinB(cosR2)ac( 2 +=β+
)BsinCcosAsinC(cosR2)ba( 2 +=γ+ [ ])CAsin()CBsin()BAsin(R2)ba()ac()cb( 2 +++++=γ++β++α+⇒ )CsinBsinA(sinR2 2 ++=
⎟⎠⎞
⎜⎝⎛ ++=
R2c
R2b
R2aR2 2
114
⇒ cba
)ba()ac()cb(R++
γ++β++α+= (2)
yk (1) + (2) eK)an rR'OC'OB'OA +=++ . 110 . bgðajtMél
PFCP;
PEBP;
PDAP
eyIgman : 1PDAD
PDDPAD
PDAP
−=−
=
Et PBC
ABC
SS
PDAD
= ,RsaydUcKñaEdrcMeBaH PEBP,
PFCP
3SS
SS
SS
PEBP
PFCP
PDAD
APC
ABC
PBA
ABC
PBC
ABC −++=++⇒
= 3S
1S
1S
1SAPCPBAPBC
ABC −⎟⎟⎠
⎞⎜⎜⎝
⎛++
= ( ) 6393S
1S
1S
1SSSAPCPBAPBC
APCPBAPBC =−≥−⎟⎟⎠
⎞⎜⎜⎝
⎛++++
⇒≥++⇒ 6PEBP
PFCP
PDAP ya:gehacNas;manmYykñúgcMenam
PEBP,
PFCP,
PDAP
mantMélelIsBI 2. ]bmafa 21
SS2
PFCP;2
PDAP
PBC
ABC >−⇒>> PBAABCPBCABC S3S;S3S >>⇔
( )( )
PEBP
SS12
SS3SS3
S3S3S2SSS3S3S2
SS3S2
PAC
ABC
PAC
ABCABCPAC
ABCPACABC
APCPBAPBCPACABC
PBAPBCABC
=+−>⇒>⇒>⇒
>+⇒++>+⇒
+>⇒
dUcenH ya:gehacNas;manmYykñúgcMenam PEBP,
PFCP,
PDAP mantMéltUcCag 2 .
115
111> KNnaGgát;p©iténrgVg; tag H CaeCIgcMenalEkgBI C eTA AB
D CaeCIgcMenalEkgBI O eTA BC
r CakaMrgVg; nig BAC=α
kñúgrgVg;p©it O man BOD2
BOC==α (1)
ΔODB CaRtIekaNEkgman
r5
OB2CB
OBDBBODsin === (2)
ΔACH CaRtIekaNEkgman 32
64
ACCHsin ===α (3)
tam (1), (2) & (3) ⇒ 32
r5= ⇒
215r = .
112> RsaybBa¢ak;fa ABCD CactuekaNcarwkeRkArgVg; yk CaRbEvgén AB, BC, t,z,y,x,d,c,b,a
CD, OA, OB, OC & OD erogKña & COB=α
ΔOAB man 2
ayxBOAsinxy
pSrprS 11 ++==⇒=
Et α−= º180BOA
⇒ α++
=⇔++α
=sinxy
ayxr1
2ayx
sinxyr1
1
116
RsaydUcKñaEdr ⇒ α++
=α++
=α++
=sinxt
dtxr1;
sinztctz
r1;
sinyzbzy
r1
432
⇒α++
+α++
=α++
+α++
sinxtdtx
sinyzbzy
sinztctz
sinxyayx
⇔ xtd
t1
x1
yzb
z1
y1
ztc
t1
z1
xya
y1
x1
+++++=+++++
⇔ xtd
yzb
ztc
xya
+=+
⇔ dyzbxtcyxazt +=+ ⇔ 22 )dyzbxt()cyxazt( +=+
⇔ bdxyzt2zydtxbacxyzt2xyctza 222222222222 ++=++
⇔ acxyztyx)coszt2tz(tz)cosxy2yx( 22222222 +α+++α++
bdxyzt2zy)cosxt2tx(tx)cosyz2zy( 22222222 +α−++α−+=
⇔ bd2cosyz2cosxt2ac2cosxy2coszt2 +α−α−=+α+α ⇔ bd2)zyb()txd(ac2)yxa()tzc( 222222222222 +−−+−−=+−−+−−
⇔ 22 )db()ca( +=+
⇔ dbca +=+
dUcenH ABCD CactuekaNcarwkeRkArgVg; . 113> RsayfatMélpleFob T minGaRs½ynwgrUbragrbs;RtIekaN eyIgbBa©Úl ABC eTAkñúgtMruymanKl; A ehIy AB sßitenAelIG½kSGab;sIus kMnt;yk )z4,y4(C&)0,x4(B
eday D, E, F CacMnuckNþal BC, CA, AB nig P, Q, RCacMnuckNþal énemdüan AD, BE, CF eK)an³ )z,y2x(R&)z,yx2(Q),z,yx(P),0,x2(F),z2,y2(E),z2,y2x2(D ++++ bNþakaerénGgát;nImYy²KW
117
222222 zyxy4x4z)yx2(AQ +++=++=
222222 z4y4xy4xz4)y2x(AR +++=++=
222222 zyxy6x9z)yx3(BP ++−=+−=
222222 z4y4xy12x9z4)y2x3(BR ++−=+−=
222222 z9y9xy6xz9)y3x(CP ++−=+−=
222222 z9y9xy12x4z9)y3x2(CQ ++−=+−=
enaHplbUksrubénbNþaGgát;TaMgenaHKW (*) )zyxyx(28 222 ++−
kaerRbEvgRCugén ΔABC KW , , 22 x16AB = 222 y16x16CA +=
2222 z16y16xy32x16BC ++−=
eK)an (**) )zyxyx(32CABCAB 222222 ++−=++
⇒ pleFob 87
(**)(*)T == efr
dUcenHtMélpleFob T minGaRs½ynwgrUbragrbs;RtIekaN . 114> bgðajfamancMnuc S mYyenAelI PR Edl PS & QS mantMélKt; tag PS = x nig QS = y ( Ν∗∈y,x )
tamRTwsIþbT cosin kñúg ΔRPQ eyIg)an³ 5.0QPRcosQPRcos.8.15.281513 222 =⇒−+=
Binitükñúúg ΔSPQ: QPRcos8.x.28xy 222 −+=
⇒ )15x0(64x8xy 2 <<++=
yk eTACMnYskñúgsmIkar 14....,,3,2,1x =
eK)anKUEdlCacMnYnKt;KW (3, 7), (5, 7) & (8, 8)
118
dUcenHmancMnuc S EdlsßitenAelI PR Edl PS & SQ manRbEvgKt;. 115> RsaybBa¢ak;fa MEMCMDMBMA +=++ tag R CakaMrgVg; (O) nig )
5x0(x2 π<< CargVas;mMu FñÚ AM .
tamRTwsþIbT sin eK)an ; xsinR2MA =
⎟⎠⎞
⎜⎝⎛ −π
= x5
sinR2MB ;
⎟⎠⎞
⎜⎝⎛ +
π= x
52sinR2MD ;
⎟⎠⎞
⎜⎝⎛ −
π= x
52sinR2MC ;
⎟⎠⎞
⎜⎝⎛ +π
= x5
sinR2ME
+ ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +π
+⎟⎠⎞
⎜⎝⎛ −
π=+ x
5sinx
52sinR2MEMC
⎟⎠⎞
⎜⎝⎛ −ππ
=⎟⎠⎞
⎜⎝⎛ −ππ
= x10
cos.5
cosR4x10
cos.103sinR2
+ ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +
π+⎟⎠
⎞⎜⎝⎛ −π
+=++ x5
2sinx5
sinxsinR2MDMBMA
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +ππ
+= x10
cos.103sin2xsinR2
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +ππ
+= x10
cos5
cos2xsinR2
ebI MEMCMDMBMA +=++
119
⇔ ⎟⎠⎞
⎜⎝⎛ −ππ
=⎟⎠⎞
⎜⎝⎛ +ππ
+ x10
cos5
cos2x10
cos5
cos2xsin
⇔ ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +π
−⎟⎠⎞
⎜⎝⎛ −ππ
= x10
cosx10
cos5
cos2xsin
⇔ xsin5
2cos.5
cos4xsin.10
sin.5
cos4xsin ππ=
ππ= (*)
eyIgeFVIkarKNna 5
2cos
5cos
ππ
eyIgman 5
3cos
52
cosπ
=⎟⎠⎞
⎜⎝⎛ π
−π
⇔ 5
3cos
52
cosπ
=π
−
⇔ 5
cos35
cos45
cos21 32 π−
π=
π−
⇔ 015
cos35
cos25
cos4 23 =−π
−π
+π
tag 5
cost π=
eK)an 01t3t2t4 23 =−−+
⇔ 0)1t2t4)(1t( 2 =−−+
eday 1t −≠
⇒ 01t2t4 2 =−−
4
15t1+
= ; 04
15t2 <+−
= minyk
eK)an 4
155
cos+
=π
4
)15(1
165215
.215
cos25
2cos 2 −
=−++
=−π
=π
enaH 41
52cos
5cos =
ππ
120
tam (*) eK)an xsinxsin41
.4xsin == Bit dUcenH MEMCMDMBMA +=++ . 116> Rsayfa RPQR&º90PRQ == tamRTwsþIbT sin kñúg ΔAQC:
º15cos2
bº105sin2
bAQ ==
dUcKñaEdr º15cos2
cRBAR;º15cos2
aPB === tamRTwsþIbT cosin:
º15cos4
)º60Bcos(ac2caRB.PB2
)º60B(COS.RB.PB2RBPBRP 2
22222 +−+
=+−+
=
º15cos4
)º60Bcos(ac2caRQ 2
222 +−+=
+ Rsayfa RP =RQ
⇔ º15cos4
)º60Bcos(ac2caº15cos4
)º60Bcos(ac2ca2
22
2
22 +−+=
+−+
⇔ )º60Acos(bc2b)º60Bcos(ac2a 22 +−=+−
⇔ Asinbc3AcosbcbBsinac3Bcosaca 22 +−=+−
tamRTwsþIbTcMenal kúñg ΔABC ³ CcosabAcosbcbCcosaAcoscbCcosaAcoscb 2 =−⇔=−⇔+=
CcosabBcosacaCcosbBcoscaCcosbBcosca 2 =−⇔=−⇔+=
tamRTwsþIbT sin kúñg ΔABC:
121
Asinbc3Bsinac3Bsin
bAsin
a=⇒=
eK)an Asinbc3AcosbcbBsinac3Bcosaca 22 +−=+− Bit . naMeGay RP =RQ . + Rsayfa luHRtaEt QRP CaRtIekaNEkg ⇒ º90PRQ = 22 RP2PQ =
eyIgman º15cos2
ACAQ =
edayKNna tamrUbmnþ º15cos2
12coscos
+α=α
⇒ 2
AC)26(AQ
426
º15cos−
=⇒+
= ; 2
BC)26(BP
−= ;
2AQQC = ; 2BPPC = ; 2
AB)26(RBRA
−==
tamRTwsþIbT sin: + )º60Acos(AQ.AR2AQARRQ 222 +−+=
)º60Acos().32(AC.AB2)32(AC)32(AB 22 +−−−+−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−−+−= Asin
23Acos
21AC.AB2ACAB)32( 22
[ ]S3)AcosAC.AB2ACAB(ACAB2
)32( 2222 +−+++−
=
)S3cba(2
32 222 +++−
=
+ RsaydUcKñaEdr ⇒ )S3cba)(32(PQ 2222 +++−=
eK)an 22 RP2PQ =
naMeGay º90PRQ = .
122
117> RsaybBa¢ak;fa A34cba 222 ≥++ tamrUbmnþ Heron:
2acb.
2bca.
2cba.
2cbaA −+−+−+++
=
3
3)acb()bca()cba(.
2cbaA4 ⎥⎦
⎤⎢⎣⎡ −++−++−+++
≤
33
)cba(A42++
≤
ac2bc2ab2cba)cba( 2222 +++++=++
(*)
eday )ca()cb()ba(cba 222222222 ++++++++≤
⇒ )cba(3)cba( 2222 ++≤++
tam (*) eK)an 33
)cba(333
)cba(A42222 ++
≤++
≤
⇒ A34cba 222 ≥++ . 118> bgðajfacMgayrvagp©iténrgVg;TaMgBIrKW )r2R(R − tag ABC CaRtIekaNsm)atEdlman AB = AC. O & I Cap©itrgVg;carwkeRkA & carwkkñúg
ΔABC. d CacMgayrvagp©itTaMg2. M CaeCIgkMBs;BI A eTAelI BC. H CaeCIgkMBs;BI A eTAelI AB. tag ⇒ BAO=α
dRr
OIAOHI
AIHIsin
+=
+==α (*)
müa:geTotR
drOB
IMOIOBOMMOBcos2cos +
=+
===α (**)
123
yk (*), (**) CMnYscUlrUbmnþ rYcTajCaplKuNktþa α−=α 2sin212cos
eK)an [ ] 0)r2R(Rd)rRd( 2 =−−++ eday ⇒ 0)rRd( >++ )r2R(Rd0)r2R(Rd2 −=⇔=−−
dUcenHcMgayrvagp©iténrgVg;TaMgBIrKW )r2R(R − .
119> KNnaplbUksrubrvagRklaépÞrgVg;TaMg4
tag r1, r2, r3, r4 CaRbEvgkaMrgVg;carwkkñúgRtIekaNTaMg 4 nig S CaRklaépÞén ΔABC. eyIg)an³
pSr1 = & )cp)(bp)(ap(pS −−−=
bNþabnÞat;b:HTaMg3RsbnwgRCugTaMg3én ΔABC ⇒RtIekaNTaMg3 dUcnwg ΔABC
EdleyIgTaj)an³
c
4
1
4
b
3
1
3
a
2
1
2
rh
rr
;hh
rr
;hh
rr
===
Edl CakMBs;EdlKUsecjBIkMBUl A, B, C énRtIekaNtUcTaMg3. )4,3,2i(h i =
cMgayrvagRCugénRtIekaNtUcTaMg3eTAnwgRCugQmén ΔABC KW 2r1 . naMeGay 1c41b31a2 r2hh;r2hh;r2hh −=−=−= kñúg ΔABC man
cS2h;
bS2h;
aS2h cba ===
enaH 22
2
a
21
12a
1
a
1a
1
2
p)ap(S
S2.pa.S.2
pS
hr2
rrhr2
1h
r2hrr −
=−=−=⇒−=−
=
eFVIdUcKñaEdr eK)an 2423 p)cp(Sr;
p)bp(sr −
=−
=
124
tag A CaplbUkRklaépÞsrubénrgVg;TaMg 4
⇒ )rrrr(A 24
23
22
21 +++π=
⎥⎦
⎤⎢⎣
⎡ −+
−+
−+π= 4
22
4
22
4
22
2
2
p)cp(S
p)bp(S
p)ap(S
pS
[ ]22224
2
)cp()bp()ap(ppS
−+−+−+π=
3
222
)cba()acb)(bca)(cba)(cba(A
++−+−+−+++π
= .
CMBUk 4 lMhat;RtiHriH
01> kñúg ΔABC man AB = AC . rgVg;mYyb:HxagkñúgénrgVg;carwkeRkAénRtIekaN nig b:HnwgRCug AB, AC Rtg; P & Q erogKña. bgðajfacMnuckNþalén PQ Cap©itrgVg; carwkkñúgénRtIekaN.
02> eKeGay A CacMnucmYykñúgcMenamcMnuc®bsBVBIrénrgVg;BIrEdlminb:unKña (C1) &
(C2) . bnÞat;mYykñúgcMenambnÞat;Edlb:HeTAnwgrgVg;TaMgBIrb:HrgVg; (C1) Rtg; P1
nig (C2) Rtg; P2 xN³EdlbnÞat;mYyeTotb:H (C1) Rtg; Q1 nig (C2) Rtg; Q2. 125
eKeGay M1 CacMnuckNþalén P1Q1 nig M2 CacMnuckNþalén P2Q2. RsaybBa¢ak;fa . 2121 MAMOAO =
03> rgVg;p©it O kat;tamkMBUl A & C énRtIekaN ABC nig kat;RCug AB & AC Rtg; K & N erogKña. rgVg;carwkeRkARtIekaN ABC nig KBN kat;KñaRtg;BIrcMnuc B &
M epSgKña. RsaybBa¢ak;fa ∠OMB CamMuEkg.
04> eKeGay ABCD CactuekaNe)a:gEdlepÞogpÞat;l½kçxNг a. BCADAB +=
b. mancMnuc P kñúgctuekaNEdlmancMgay h BI CD Edl BChBP&ADhAP +=+= RsaybBa¢ak;fa
BC1
AD1
h1
+= .
05> Ggát;FñÚEdlmanRbEvg 3 EckrgVg;EdlmankaMÉktaCaBIrtMbn;. kMnt;ctuekaNEkgEdlmanRklaépÞbMputEdlGaccarwkkñúgtMbn;tUc.
06> Ggát;FñÚ AB & CD énrgVg;mYykat;KñaRtg;cMnuc E EdlsßitkñúgrgVg;. eKeGay M
Ca cMnucenAelIGgát; EB. bnÞat;b:HRtg; E eTAnwgrgVg;Edlkat;tam D, E, M kat; bnÞat; BC & AC Rtg; F & G erogKña. KNna
EFEG CaGnuKmn_
ABAMt = .
07> eKeGay I Cap©itRtIekaN ABC. rgVg;carwkkñúg ΔABC b:HRCug BC, CA & AB
Rtg; K, L, & M erogKña. bnÞat;kat;tam B ehIyRsb MK kat; LM & LK
Rtg; R & S erogKña. bgðajfa ∠RIS CamuMRsYc.
126
08> bnÞat; AB b:HeTAnwgrgVg; CAMN & NMBD. M enAelI CD
ehIysßitenAcenøaH C & D nig CD Rsbnwg AB. Ggát;FñÚ NA & CM
kat;KñaRtg; P ehIyGgát;FñÚ NB & MD kat;Kña®tg; Q. bnøay CA & DB
kat;KñaRtg; E. Rsayfa QEPE = .
09> eKeGay ABC CaRtIekaNEdlman . D & E Ca cMnucenAelI AC & AB Edl . BD kat; CE Rtg; F. bgðajfabnÞat; AF Ekgnwg bnÞat; BC.
º60CBA&º40CAB ==
º70ECB&º40DBC ==
10> eKeGayctuekaNe)a:g ABCD Edlman nig AB=CB. bgðajfa AD = CD.
BDC2DBA,BDA2DBC ==
11> ABC CaRtIekaNEkg®tg; C. knøHbnÞat;BuHmMukñúgén ∠BAC & ∠ABC kat; BC
& CA Rtg; P & Q erogKña. cMnuc M & N CaeCIgcMenalEkgBI P & Q eTAelI AB. KNna ∠MCN.
12> eKeGayctuekaN ABCD carwkkñúgrgVg;kaM r. Ggát;RTUg AC & BD kat;KñaRtg; E. bgðajfaebI AC⊥BD enaH . 22222 r4EDECEBEA =+++
13> Ggát;RTUgénctuekaNe)a:g ABCD RbsBVKñaRtg; O. p©itén ΔAOD & ΔBOC KW P & Q . GrtUsg;én ΔAOB & ΔCOD KW R & S. bgðajfa PQ ⊥ RS .
14> RtIekaN ABC manGrtUsg; H. eCIgcMenalEkgén H elIknøHnÞat;BuHmMukñúg nig eRkAén ∠BAC KW P & Q. bgðajfa PQ kat;tamcMnuckNþalén BC.
127
15> P, Q & R CacMnucenAelI RCug BA, CA & AB erogKñaén ΔABC. bgðajfa RtIekaNEdlmankMBUlCap©iténrgVg;carwkeRkA ΔAQR, ΔBRP & ΔCPQ dUcnwg ΔABC .
16> eKGay ΔABC Edlman AB = AC & O Cap©itrgVg;carwkeRkA. D CacMnuc kNþal AB nig E CaTIRbCMuTMgn;én ΔACD. bgðajfa OE ⊥ CD.
17> ctuekaNe)a:g PQRS manRklaépÞ A. O CacMnucmYyenAkñúg PQRS. Rsayfa ebI enaH PQRS CakaerEdlman O Cap©it. 22222 OSOROQOPA2 +++=
18> cUrrkRKb; ΔABC Edl 5BDAD&2ACAB =+=+ . AD CakMBs;KUs ecjBI A kat; BC Rtg; D.
19> RCug BC, CA & AB énRtIekaNb:HnwgrgVg;Rtg; X, Y & Z. Rsayfap©iténrgVg; sßitenAelIbnÞat;Edlkat;tamcMnuckNþalén BC nig cMnuckNþalén BC AX.
20> kMnt;RbEvgénRCug ΔABC EdlRbEvgénkMBs;nImYy²KW 3, 4, 6 .
128
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