CMBUk 1 segçbrUbmnþsMxan;;² 1> TMnak;TMngkñúgRtIekaN · 2 A cos 2 C cos 2 B acos 2 A a r ptan...

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 ≤ .


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 ≥++ .


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


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

++≥++ .


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;l½kçxNÐ³ 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

CMBUk 1 segçbrUbmnþsMxan;;² 1> TMnak;TMngkñúgRtIekaN · 2 A cos 2 C cos 2 B acos 2 A a r ptan...

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Transcript of CMBUk 1 segçbrUbmnþsMxan;;² 1> TMnak;TMngkñúgRtIekaN · 2 A cos 2 C cos 2 B acos 2 A a r ptan...