Eulclide Geometry.pdf

A

B C D M

N

Q

P E

F

- i -

KNkmkarniBnnigeroberog

elak lwm pln nig elak Esn Bisid

KNkmkarRtYtBinit

elak lwm qun elak Gwug sMNag elak Tit em:g GkRsI Tuy rINa elak RBwm sunit elak pl bunqay elak elak nn; suxNa

GkRtYtBinitGkraviruT

elak lwm miKsir

karIkMuBTr GkrcnaRkb kBaa lI KuNaka elak lwm pln

- ii -

Garmfa esovePA sikSaKNitviTaedayxng Epk FrNImaRtGWKIt EdlGksikSakMBugkan;sikSaenAkgdenH eyIgxJM)anxitxMRsavRCavcgRkgeLIg kgeKalbMNgTukCaksa sikSabEnmelIemeroncMnYnkMupicedayxng . enAkgesovePAenHrYmmanbICMBUkKW CMBUkTI1 emeronsegbPab;CamYy]TahrN_KMrU CMBUkTI2 lMhat;eRCIserIsnigdMeNaHRsay nig CMBUkTI3 CalMhat;Gnuvtn_ . esovePAenHminlhYseK hYsgenaHeT kMhusedayGectnaR)akdCaman GaRsyehtuenH eyIgxJMCaGkniBn nig eroberog rgcaMCanicnUvmtiriHKn;BIsMNak; GksikSakgRKb;mCdanedaykIrIkray edIm,IEklMGesovePAenH[kan;Etman suRkitPaBbEnmeTot . CaTIbBab;eyIgxJMsUmCUnBrcMeBaHGksikSaTaMgGs;CYbEtsuPmgl suxPaBl nigTTYlCyCMnkgkarsikSa nig muxrbrkargar RKb;eBlevla .

tdMbgfTI 30 mIna 2011 GkniBn lwm pln Tel : 017 768 246 www.mathtoday.wordpres.com

- iii -

lwm pln nig Esn Bisid

rkasiTiRKbyag

FrNImaRtviPaKkgbg;

eroberogeday lwm pln TMBr 1

CMBUkTI1

RTwsIbTsMxan;

1-RTwsIbTkUsIunUs ( The law of sines ) cMeBaHRtIekaN ABC manRCug cAB,bAC,aBC carwkkgrgVg;pit O kaM R eK)an R2

Csinc

Bsinb

Asina .

sRmaybBaak; sg;Ggt;pit R2BD enaHeK)an o90BAD mMucarwkknHrgVg; kgRtIekaNEkg ABD eKman

R2c

BDABDsin

eday ACBADB mMucarwkkgrgVg;sat;edayFrrYm AB

A

B

C

DO

FrNImaRtviPaKkgbg;


eK)an R2cCsin b R2

Csinc .

RsaydUcKaEdreK)an R2Asin

a nig R2Bsin

b dUcenH R2

Csinc

Bsinb

Asina .

RTwsIbTenHBitCaniccMeBaHmMu C,B,A CamMuRsYc b mMuEkg b mMuTal . 2-rUbmncMeNalEkg (Projection Formula ) kgRKb;RtIekaN ABC eKman

AcosbBcosacCcosaAcoscbBcoscCcosba

sRmaybBaak; sg;km

FrNImaRtviPaKkgbg;


eKman HCBHBC kgRtIekaNEkg ABH nig AHC eKman

cBH

ABBHBcos enaH BcoscBH

bHC

ACHCCcos enaH CcosbHC

ehtuenH CcosbBcoscBC eday aBC dUcenH BcoscCcosba . cMeBaHrUbmnBIreTotRsaydUcrebobxagelIenHEdr . 3-RTwsIbTkUsIunUs kgRKb;RtIekaN ABC eKman

Ccosab2bac

Bcosca2acb

Acosbc2cba

222

222

222

sRmaybBaak; tamrUbmncMeNalEkgeKman

AcosbBcosacCcosaAcoscbBcoscCcosba

b

AcosbcBcosacc

CcosabAcosbcb

BcosacCcosaba

2

2

2

eFIVplbUksmIkarBIrenHGg nig GgeK)an

FrNImaRtviPaKkgbg;


Acosbc2cba 222 b Acosbc2cba 222 cMeBaHrUbmnBIreTotRsaydUcKaxagelIEdr . 4-RTwsIbTemdan kgRKb;RtIekaN ABC EdlmanRCug c,b,a nigemdanRtUvKa cba m,m,m eKman

4c

2bam

4b

2acm

4a

2cbm

2222

c

2222

b

2222

a

sRmaybBaak; sg;emdan amAK

A

B C K

FrNImaRtviPaKkgbg;


tamRTwsIbTkUsIunUskgRtIekaN ABK BcosBK.AB2BKABAK 222 b )1(Bcosac

4acBcos

2a.c2

4acm

22

222

a tamRTwsIbTkUsIunUskgRtIekaN AKC CcosKC.AC2KCACAK 222 b )2(Ccosab

4abCcos

2a.b2

4abm

22

222

b bUksmIkar )2(&)1( eK)an

)CcosbBcosc(a2

acbm22

222a

Et aCcosbBcosc rUbmncMeNalEkg enaH

2acba

2acbm

2222

2222

a

dUcenH 4

a2

cbm222

2a .

4-RTwsIbTLibnic Leibnizs theorem kgRKb;RtIekaN ABC EdlmanRCug c,b,a ehIy O CapitrgVg;carwkkg nig G CaTIRbCMuTmn;nRtIekaNenaHeKman

9cbaROG

22222 .

FrNImaRtviPaKkgbg;


sRmaybBaak;

sg;dan AK nig BL kat;KaRtg; G . tag OGA enaH OGK tamRTwsIbTkUsIunUskgRtIekaN OGA

cosGA.OG2GAOGOA 222 eKTaj )1(

GA.OG2OAGAOGcos

222 tamRTwsIbTkUsIunUskgRtIekaN OGK

cosGK.OG2GKOGOK

)cos(GK.OG2GKOGOK222

22

A

B C K

G O

FrNImaRtviPaKkgbg;


eKTaj)an )2(GK.OG2

OGGKOKcos222

pwm )1( nig )2( eK)an

)OGGKOK(GKGAOAGAOG

GK.OG2OGGKOK

GA.OG2OAGAOG

222222

222222

edayeKman aa m31AK

31GK,m

32AK

32GA,ROA

kgRtIekaNEkg :OCK 4

aROK2

22 BItaKr

eK)an )OGm91

4aR(2Rm

94OG 22a

2222

a2

b 2

am32R3OG3

22

a22

tamRTwsIbTemdan 4

a2

cbm222

2a

ehtuenH 2

a)4

a2

cb(32R3OG3

222222

3

cbaR3OG3222

22 dUcenH

9cbaROG

22222 .

FrNImaRtviPaKkgbg;


5-RTwsIbT Stewart Stewart's Theorem eK[RtIekaN ABC mYymanRCug c,b,a . P CacMNucmYyn AB Edl nPB,mPA nig cnm . eK)an 22222 nmmnPC).nm(nbma . smal;eKehARTwsIbT Stewart dUcKanwgRTwsIbT Apollonius (Apollonius' theorem). sRmaybBaak; tamRTwsIbTkUsIunUscMeBaHRtIekaN PAC nig PBC eKman

cosPA.PC2PAPCAC 222 Edl APC cos.PB.PC2PBPCBC 222

eday nPB,mPA,aBC,bAC

C

A BPm n

FrNImaRtviPaKkgbg;


eK)an cos.mPC2mPCb 222 nig cosnPC2nPCa 222 ehtuenH 22222 mnnmPC)nm(manb dUcenH 22222 nmmnPC).nm(nbma . 6-pRkLanRtIekaN k>krNIsal;RCug nig km

FrNImaRtviPaKkgbg;


eKman aha21BC.AH

21S

kgRtIekaNEkg AHC eKman bh

ACAHCsin a b Csinbha

dUcenH Csinab21S .

smal; tamRTwsIbTsIunUseKman

R2cCsin Edl R CakaMrgVg;carwkeRkARtIekaN

ehtuenH R4

abcR2c.ab

21Csinab

21S .

dUcenH R4

abcCsinab21Bsinca

21Asinbc

21S .

K>rUbmnehr:ug RTwsIbT pRkLan ABC edaysal;RCugTaMgbI c,b,a kMNt;eday )cp)(bp)(ap(pS Edl

2cbap

sRmaybBaak; tamrUbbmn )1(Asinbc

21S

eKman Acos1Asin 22 eday Acosbc2cba 222 enaH

bc2acbAcos

222

FrNImaRtviPaKkgbg;


eK)an 2222

2 )bc2

acb(1Asin

22

22

2222

22

222222

cb4)cba)(cba)(acb)(acb(

cb4])cb(a][a)cb[(

cb4)acb(cb4

tag 2

cbap CaknHbrimaRtn ABC

enaHeK)an

)bp(2cba)cp(2cba)ap(2acb

p2cba

eK)an 222

cb4)cp)(bp)(ap(p16Asin

eKTaj )2()cp)(bp)(ap(pbc2Asin

yk )2( CYskg )1( eK)an )cp)(bp)(ap(p

bc2bc

21S

dUcenH )cp)(bp)(ap(pS .

FrNImaRtviPaKkgbg;


X>RkLapRtIekaNCanuKmn_nknHbrimaRt nig kaMrgVg;carwkkg RTwsIbT pRkLan ABC Edlman p CaknHbrimaRt nig r CakaM nrgVg;carwkgnRtIekaNkMNt;eday prS . sRmaybBaak; tag I CapitrgVg;carwkkgRtIekaN ABC ehIy L,K,H CacMnucb:Hrvag rgVg;carwkkgRtIekaN CamYyRCug AB,CA,BC erogKa . pRkLan ABC kMNt;eday r.

2cbacr

21br

21ar

21SSSS IABICAIBC

tag

2cbap enaH prS .

H B C

A

KL

I

FrNImaRtviPaKkgbg;


7-kaMrgVg;carwkkg nig kaMrgVg;carwkeRkAnRtIekaN eK[RtIekaN ABC mYymanRCug c,b,a ehIytag

2cbap .

tag r nig R erogKaCakaMrgVg;carwkkg nig kaMrgVg;carwkeRkAn ABC tamrUbmnpRkLanRtIekaN ABC eK)an )cp)(bp)(ap(p

R4abcprS

dUcenH p

)cp)(bp)(ap(r

nig )cp)(bp)(ap(p4

abcR . 8-TMnak;TMngqHnRCugrbs;RtIekaNmYy eK[RtIekaN ABC mYymanRCug c,b,a . RbEvgRCug c,b,a manTMnak;TMngqHCaGnuKmn_n p,R,r dUcxageRkam

)3(prR4abc)2(rR4rpcabcab

)1(p2cba22

)annyfa c,b,a CabsnsmIkardWeRkTIbI 0prR4x)rR4rp(px2x 2223 tamRTwsIbTEvt .

cMeBaHTMnak;TMng )2( nig )3( GacRsaybBaak;dUcxageRkam

FrNImaRtviPaKkgbg;


tamrUbmn p

)cp)(bp)(ap(r

eK)an p

)cp)(bp)(ap(r2

p

abcp)cabcab(p)cba(pr23

2

eday p2cba ehIy prR4

abcS enaH prR4abc

ehtuenH p

prR4p)cabcab(p2pr33

2

rR4cabcabpr 22 dUcenH rR4rpcabcab 22 . smal; eKman ca2bc2ab2cba)cba( 2222 eK)an )cabcab(2)cba(cba 2222

rR8r2p2

)rR4rp(2)p2(22

222

dUcenH )4(rR8r2p2cba 22222 ehIy abc)acbcab)(cba()ac)(cb)(ba(

)rR2rp(p2

prR4)rR4rp(p222

22

FrNImaRtviPaKkgbg;


dUcenH )5()rR2rp(p2)ca)(cb)(ba( 22 tamsmPaB )ca)(cb)(ba(3cba)cba( 3333 eK)an )ca)(cb)(ba(3)cba(cba 3333

)rR6r3p(p2

)rR6r3p3p4(p2

)rR2rp(p6)p2(

22

222

223

dUcenH )6()prR6pr3p(2cba 23333 ]TahrN _ kgRKb;RtIekaN ABC cUrRsayfa

Rr1CcosBcosAcos Edl r nig R tagerogKa

CargVas;kaMrgVg;carwkkg nig kaMrgVg;carwkeRkAnRtIekaN ABC . dMeNaHRsay tamRTwsIbTkUsIunUsGnuvtn_kgRtIekaN ABC eK)an

ab2cba

ac2bac

bc2acbCcosBcosAcos

222222222

abc2

)cba(2)cba)(cba( 333222 edayCMnYs p2cba ehIy rR8r2p2cba 22222 nig )prR6pr3p(2cba 23333

FrNImaRtviPaKkgbg;


eK)an

prR8)prR6pr3p(4)rR8r2p2(p2CcosBcosAcos

2322

Rr1

rR4r4rR4

pRr8)rR12r6p2rR8r2p2(p2

2

2222

dUcenH Rr1CcosBcosAcos .

9-RTwsIbTGWEl eK[RtIekaN ABC mYymanRCug c,b,a . tag I CapitnrgVg;carwk kgnRtIekaNenH . cMeBaHRKb;cMnuc X nbg;eKmanTMnak;TMng

abcXI.)cba(XC.cXB.bXA.a 2222 . sRmaybBaak;

B

A

C

I

y

x

FrNImaRtviPaKkgbg;


kgtRmuyGrtUnrm:al; )Bxy( eKman )Bsinc;Bcosc(A;)0,a(C;)0;0(B .

tag r CakaMrgVg;carwkkg nig 2

cbap CaknHbrimaRt nRtIekaN ABC enaHeK)an )r;bp(I . ebI S CapRkLanRtIekaN ABC enaHeKman

pr)cp)(bp)(ap(pS rUbmnehr:ug elIkCakaereKTaj)an

p)cp)(bp)(ap(r2

tag )y;x(X 00 CacMnucNakedaynbg; . tag 222 XC.cXB.bXA.aM nig abcXI).cba(N 2 eK)an

2200

20

20

2200

20

20

220

222

02

02

0

2200

20

20

20

20

20

20

20

20

222

accaSy4)bp(px4)yx(p2

accaSy4)b2p2)(p2(x)yx(p2

accaSy4)ac2

bca1(acx2)yx(p2

caacBsinacy2)Bcos1(acx2)yx)(cba(

]y)ax[(c)yx(b])Bsincy()Bcoscx([a

cXCbXBaXAM

FrNImaRtviPaKkgbg;


ehIy abcXI)cba(N 2

abc)cp)(bp)(ap(2)bp(p2Sy4)bp(px4)yx(p2

abcp

)cp)(bp)(ap(p2

)bp(p2Sy4)bp(px4)yx(p2

abcpr2

)bp(p2pry4)bp(px4)yx(p2

abc)ry())bp(x(p2

200

20

20

200

20

20

2

200

20

20

20

20

yk abc)cp)(bp)(ap(2)bp(p2T 2

22

2

22

acca

)ca(ac)bb2p2(acabc)bp(ac2abc]acp)cba(p2)[bp(2

abc]acp)ca(ppbp)[bp(2

eKTaj 22002020 accaSy4)bp(px4)yx(p2N edaykenSam NM naM[eKTaj)an

abcXI)cba(XC.cXB.bXA.a 2222 .

FrNImaRtviPaKkgbg;


10-cmayrvagpitrgVg;carwkkg nig pitrgVg;carwkeRkAnRtIekaN RTwsIbT eK[RtIekaN ABC mYyEdl O CapitrgVg;carwkeRkA nig I CapitrgVg;carwk kg . ebI R nig r CargVas;kaMrgVg;carwkkg nig kaMrgVg;carwkeRkAnRtIekaNenaH eK)an Rr2ROI 22 b )r2R(ROId . sRmaybBaak; tamRTwsIbTGWEleyIg)an

abcOI)cba(OC.cOB.bOA.a 2222 eday ROCOBOA nig

2cbap

eK)an abcd.p2Rp2 22 naM[ p2

abcRd 22 Edl OId .

tamrUbmnpRkTLa R4

abcprS eKTaj rR2p2

abc dUcenH )r2R(RrR2Rd 22 b )r2R(Rd .

FrNImaRtviPaKkgbg;


11-cmayBIpitrgVg;carwkkgeTAkMBUlnRtIekaN eK[RtIekaN ABC mYymanRCug c,b,a . ebI I CapitrgVg;carwkkgnRtIekaNenaHeK)an

ab.p

cpIC,ac.p

bpIB,bc.p

apIA sRmaybBaak; tamRTwsIbTGWElcMeBaHRKb;cMnuc X eKman

abcXI).cba(XC.cXB.bXA.a 2222 ebI AX enaHeK)an

abcAI).cba(cbbc

abcAI).cba(AC.cAB.bAA.a222

2222

A

B C

I

FrNImaRtviPaKkgbg;


eKTaj)an p2

)a2p2(bccba

)acb(bcAI2

dUcenH bc.p

apAI . cMeBaHrUbmnBIreTotRsaydUcKa . smal; eKman ab.

pcpIC,ac.

pbpIB,bc.

papIA

eK)an )p

cpp

bpp

ap(abcIC.cIB.bIA.a 222

eday 1p

p2p3p

)cba(p3p

cpp

bpp

ap dUcenH abcIC.cIB.bIA.a 222 . 12-kenSamRtIekaNmaRt

2Atan,

2Acos,

2Asin

eK[RtIekaN ABC mYymanRCug c,b,a nigknHbrimaRt p . eKmanTMnak;TMngdUcxageRkam

)cp(p

)cp)(bp(2Atan,

bc)ap(p

2Acos

bc)cp)(bp(

2Asin

FrNImaRtviPaKkgbg;


sRmaybBaak; kgRtIekaNEkg IAL eKman

IAr

IALI

2Asin

eday p

)cp)(bp)(ap(r nig bc.p

apIA

eK)an bc

)cp)(bp(

bc.p

app

)cp)(bp)(ap(

2Asin

dUcenH bc

)cp)(bp(2Asin . dUcKaEdreKmanTMnak;TMng

ac

)cp)(ap(2Bsin nig

ab)bp)(ap(

2Csin .

H B C

A

KL

I

FrNImaRtviPaKkgbg;


ma:geToteKman 2Acos

2AsinbcAsinbc

21S

eKTaj bc

)cp)(bp(bc

)cp)(bp)(ap(p

2Asinbc

S2Acos

dUcenH bc

)ap(p2Acos . RsaydUcKaEdreK)an

ac)bp(p

2Bcos nig

ab)cp(p

2Ccos .

eKman )ap(p

)cp)(bp(

bc)ap(p

bc)cp)(bp(

2Acos2Asin

2Atan

RsaydUcKaEdreK)anTMnak;TMng

)bp(p)cp)(ap(

2Btan

nig )cp(p

)bp)(ap(2Ctan

. 13-RTwsIbTknHbnat;BuHkgnmMumYyrbs;RtIekaN eK[RtIekaN C,B,A manRCug c,b,a . ebI CBA ,, CaknHbnat;BuHkgnmMu C,B,A enaHeK)an

2Ccos

baab2,

2Bcos

caac2,

2Acos

cbbc2

CBA .

FrNImaRtviPaKkgbg;


sRmaybBaak; pRkLanRtIekaN ABC KW

2Asinb

21

2Asinc

21Asinbc

21

SSS

AA

AKCABK

eKTaj)an 2Asin)cb(

AsinbcA eday

2Acos

2Asin2Asin

dUcenH 2Acos

cbbc2

A . RsaydUcKaEdreK)an

2Ccos

baab2,

2Bcos

caac2

CB .

A

K B C

A

FrNImaRtviPaKkgbg;


14-kenSamkaMrgVg;carwkkgnRtIekaNmYy eK[RtIekaN C,B,A manRCug c,b,a nigknHbrimaRt p . ebI r CakaMrgVg;carwkkgRtIekaN ABC enaHeK)an

2Ctan)cp(

2Btan)bp(

2Atan)ap(r .

sRmaybBaak; tag zCKCH,yBHBL,xAKAL

eK)an

bxzazycyx

naM;[ p2cba)zyx(2

eKTaj pzyx enaH

cp)yx(pzbp)xz(pyap)zy(px

HB C

A

KL

I

FrNImaRtviPaKkgbg;


dUcenH bpBHBL,apALAK nig cpCKCH . kgRtIekaNEkg ALI eKman

apr

ALLI

2Atan

eK)an 2Atan)ap(r . RsaydUcKaEdreK)an

2Btan)bp(r nig

2Ctan)cp(r

dUcenH 2Ctan)cp(

2Btan)bp(

2Atan)ap(r .

smal; pRkLanRtIekaN ABC GacKNnatamrUbmn

2Ctan)cp(p

2Btan)bp(p

2Atan)ap(pprS .

15-kenSamkaMrgVg;carwkkgmMumYynRtIekaN RTwsIbT eK[RtIekaN C,B,A manRCug c,b,a nigknHbrimaRt p . ebI Ar CakaMrgVg;carwkkgmMu A nRtIekaN ABC enaHeK)an

2Ctan

bp

2Btan

cp2AtanprA

.

FrNImaRtviPaKkgbg;


sRmaybBaak; eKman CHACCKACAK nig BHABBLABAL eK)an

p2BCABAC)BHCH(ABACALAK eday ALAK enaHeK)an p2AL2AK2 b pALAK ehIy bpACAKCHCK nig cpBLBH dUcenH cpBLBH,bpCHCK,pALAK kgRtIekaNEkg AL eKman

pr

ALL

2Atan A

eK)an )1(2AtanprA

A B

C

H

K

L

Ar

FrNImaRtviPaKkgbg;


m:ageToteKman LBHB naM[ BH

22LBH

22B

eK)an Ar

cpH

BHBHcot2Btan eKTaj )2(

2Btan

cprA

RsaydUcKaEdreK)an )3(2Ctan

bprA

tamTMnak;TMng )2(,)1( nig )3( eK)an

2Ctan

bp

2Btan

cp2AtanprA

.

eKmanTMnak;TMngRsedogKaenHEdrcMeBaHkaMrgVg;carwkkgmMu B nig kgmMu C

2Ctan

ap

2Atan

bp2Ctanpr,

2Ctan

ap

2Atan

cp2Btanpr CB

smal ; eKman

2Atan

bp.2Btanp.

2Btan

cp.2Atan)ap(rrrr CBA

2CBA S)cp)(bp)(ap(prrr dUcenHeK)an CBA rrrrS .

FrNImaRtviPaKkgbg;


16-RTwsIbTesv:a Cevas theorem RTwsIbT kgRtIekaN ABC mYy bnat;bI BE,AD nig CF RbsBVKaRtg;cMnuc K mYyluHRtaEt 1

EACE.

DCBD.

FBAF .

sRmaybBaak; ]bmafabnat;bI BE,AD nig CF RbsBVKaRtg;cMnucK mYy yk )( Cabnat;kat;tam A ehIyRsbnwg )BC( . tag G nig H CaRbsBVrvagbnat; )BE( nig )CF( CamYybnat; )( erogKa .

A

B C D

EF K

G H

FrNImaRtviPaKkgbg;


eday AHF dUc BCF enaHeK)an )1(BCAH

FBAF

AEG dUc BCE enaHeK)an )2(AGBC

EACE

AGK dUc BDK enaHeK)an )3(DKAK

BDAG

AHK dUc CDK enaHeK)an )4(DKAK

DCAH

tam )4(&)3( eK)an DCAH

BDAC b )5(

AHAG

DCBD

KuNTMnak;TMng )5(&)2(,)1( Gg nig GgeK)an 1

AHAG.

AGBC.

BCAH

EACE.

DCBD.

FBAF Bit . ]bmafa 1

EACE.

DCBD.

FBAF Bit.

eyIgnwgRsayfabnat; BE,AD nig CF RbsBVKaRtg;cMnuc K mYy. sntfa k CaRbsBVrvagbnat; BE nig CF ehIyyk 'D CaRbsBVrvag bnat; AK nig BC . enaHtamsRmayxagelIeK)an

1EACE.

C'D'BD.

FBAF Ettamkar]bma 1

EACE.

DCBD.

FBAF

enaHeK)an DCBD

C'D'BD b 1

DCBD1

C'D'BD

b DC

DCBDC'D

C'D'BD b DCBC

C'DBC enaH DCC'D mannyfa

cMNuc 'D nig D RtYtsIuKa .

FrNImaRtviPaKkgbg;


17-RTwsImInWLs Menelaus theorem RTwsIbT ykbIcMnuc D,F nig E siterogKaelIRCug BC,AB nig AC nRtIekaN ABC enaHcMNucbIenHrt;Rtg;KaebI 1

AECE.

CDBD.

BFAF .

sRmaybBaak;

tag cba H,H,H CacMeNalEkgn C,B,A elI )EF( erogKa . enaHeK)an FAHa dUc FBHb enaH )1(BH

AHBFAF

b

a

DBHb dUc DCHc enaH )2(CHBH

CDBD

c

b

EAHa dUc ECHc enaH )3(AHCH

AECE

a

c

KuNsmIkar )3(&)2(),1( eK)an 1AECE.

CDBD.

BFAF .

A

B C

F

D

E

bH

aH

cH

FrNImaRtviPaKkgbg;


18-sVyKuNncMnucmYyeFobnwgrgVg;mYy RTwsIbT ebIbnat;BIrKUsecjBIcMnuc P mYykat;rgVg;Rtg; B,A nig D,C erogKa enaHeK)an PDPCPBPA . sRmaybBaak; krNIcMnuc P enAkgrgVg; eKman PCBPAD mMucarwkkgrgVg;sat;edayFrYm DB CPBAPD mMuTl;kMBUl eK)an PAD nig PCB CaRtIekaNdUcKa . eK)anpleFobdMNUc

PBPD

PCPA enaH PDPCPBPA .

A B

C

D

P

FrNImaRtviPaKkgbg;


krNIcMnuc P enAeRkArgVg; eKman ABCADC mMucarwkkgrgVg;sat;FrrYm AC BPCAPD mMurYm

eK)an PAD nig PCB CaRtIekaNdUcKa . eK)anpleFobdMNUc

PBPD

PCPA enaH PDPCPBPA .

smal ; tag R CakaMrgVg; ehIy d CacmayBIpitnrgVg;eTAcMnuc P . -ebI P enAkgrgVg;enaH 22 dRPDPCPBPA -ebI P enAeRkArgVg;enaH 22 RdPDPCPBPA .

P A B

C

D

FrNImaRtviPaKkgbg;


sRmaybBaak;

krNI P enAkgrgVg; ehIyyk KL CaGgt;pitnrgVg;kat;tam P eK)an 22 dR)dR)(dR(PL.PKPB.PA .

krNI P enAeRkArgVg; ehIyyk KL CaGgt;pitnrgVg;kat;tam P eK)an 22 Rd)Rd)(Rd(PL.PKPB.PA .

A

B

P K LO

P

AB

K L O

FrNImaRtviPaKkgbg;


19-RTwsIbTtUelmI Ptolemy's theorem RTwsIbT

ebI ABCD CactuekaNcarwkkgrgVg;enaHeK)an AD.BCCD.ABBD.AC . sRmaybBaak;

yk K CacMnucmYyn AC Edl CBDABK eKmanmMucarwkkg BDCBAC nig ACBADB eK)an KBCKBDDBCKBDABKABD enaH ABK dUcRtIekaN DBC ehIy ABD dUcRtIekaN KBC eK)anpleFobdMNUc

BDCD

ABAK nig

BDDA

BCCK

A

B

C

D

K

FrNImaRtviPaKkgbg;


eKTaj CD.ABBD.AK nig AD.BCBD.CK ehtuenH AD.BCCD.ABBD.CKBD.AK AD.BCCD.AB)CKAK(BD eday ACCKAK dUcenH AD.BCCD.ABBD.AC . 20-vismPaBtUelmI Ptolemy's inequality RTwsIbT eK[ctuekaN ABCD enaHeK)an BD.ACAD.BCCD.AB vismPaBenHkayCasmPaBluHRtaEtctuekaN ABCDcarwkkgrgVg; . sRmaybBaak; yk DCBA z,z,z,z CaGahVikncMnuc D,C,B,A erogKa . eKmansmPaB

)zz)(zz()zz)(zz()zz)(zz( BDACADBCCDAB tamvismPaBRtIekaNeK)an

|)zz)(zz(||)zz)(zz(||)zz)(zz| ADBCCDABBDAc dUcenH AD.BCCD.ABBD.AC .

FrNImaRtviPaKkgbg;


21-pRkLactuekaNcarwkkgrgVg; rUbmn Brahmagupta Brahmagupta's formula eK[ctuekaN ABCD manRCug d,c,b,a nigknHbrimaRt p carwkkgrgVg;enaHpRkLavakMNt;eday )dp)(cp)(bp)(ap(S . sRmaybBaak; eday o180CA enaHeK)an AsinCsin nig AcosCcos eKman Csinbc

21Asinad

21SSS BCDABD

Asin)bcad(21S enaH )1(

bcadS2Asin

tamRTwsIbTkUsIunUsGnuvtn_kgRtIekaN ABD nig BCD

A

B

C

D

FrNImaRtviPaKkgbg;


eK)an Acosbc2cbAcosad2daBD 22222 eKTaj)an )2(

)bcad(2cbdaAcos

2222

ehIyeKman )3(1AcosAsin 22 tam )3(&)2(,)1( eK)an

1)bcad(4

)cbda()bcad(

S42

22222

2

2

eKTaj 1616

)cbda()bcad(4S222222

2 Edl )cbda()bcad(2 2222

)cbda)(cbda()cb()da( 22

nig )cbda()bcad(2 2222

)dacb)(dacb()da()cb( 22

eday p2dcba enaH )cp)(bp(4 nig )dp)(ap(4 ehtuenH )dp)(cp)(bp)(ap(S2 dUcenH )dp)(cp)(bp)(ap(S .

FrNImaRtviPaKkgbg;


smal; ebIcMnuc CD enaH 0d eK)an )cp)(bp)(ap(pS rUbmnehr:ug . 22-rUbmnpRkLactuekaNe)a:g

rUbmn Bretschneider (Bretschneider's formula)

eK[ctuekaNe)a:g ABCD manRCug d,c,b,a nigknHbrimaRt p enaHpRkLarbs;vakMNt;eday

2CAcosabcd)dp)(cp)(bp)(ap(S 2 .

A

B

C D,

FrNImaRtviPaKkgbg;


sRmaybBaak;

eKman Csinbd

21Asinad

21SSS BCDABD

b )1(S2CsinbdAsinad tamRTwsIbTkUsIunUskgRtIekaN ABD nig BCD eK)an

Ccosbc2cbAcosad2da 2222 eKTaj )2(

2cbdaCcosbcAcosad

2222 tam )2(&)1( eK)an

222 S4)CcosbcAcosad()CsinbcAsinad( 2

2222)

2cbda(

A

B C

D

a

b

c

d

FrNImaRtviPaKkgbg;


eKTaj)an

2CAcosabcd

16S

2CAcosabcd4

4)cbda(bcadS4

22

222222

22

Edl )cbda()bcad(2 2222

)cbda)(cbda()cb()da( 22

nig )cbda()bcad(2 2222

)dacb)(dacb()da()cb( 22

eday p2dcba enaH )cp)(bp(4 nig )dp)(ap(4 ehtuenH

2CAcosabcd)dp)(cp)(bp)(ap(S 22

dUcenH 2

CAcosabcd)dp)(cp)(bp)(ap(S 2 . smal ; ebI ABCD carwkkgrgVg;enaH 090cos

2CAcos o

enaH )dp)(cp)(bp)(ap(S .

FrNImaRtviPaKkgbg;


23-cmayrvagpitrgVg;carwkkg nig carwkeRkAnctuekaNe)a:g ebIctuekaNe)a:g ABCD carwkkgrgVg; )R,O(C nigcarwkeRkA rgVg; )r,I('C enaHeK)an

2222 R4rrRrOId . sRmaybBaak;

tag H nig K CacMNucb:HrvagrgVg; )'C( CamYyRCug ]AB[ nig ]BC[ ehIy E nig F CacMNucRbsBVrvag )AI( nig )CI( CamYyrgVg; )C( erogKa .

A

B C

D

I

O

H

K

E

F

FrNImaRtviPaKkgbg;


eKman DCBDCF2DOF nig DABDAE2DOE eK)an o180DABDCBDOEDOF naM[ ]EF[ CaGgt;pitnrgVg; )C( . tamRTwsIbTemdankgRtIekaN EIF eK)an

)1(R)IFIE(21

4EF

2IFIEOI 222

2222

eKman 2

BADIAB nig 2

DCBICB

eK)an oo

902

1802

DCBBADICBIAB sg; rIKIH . BIrRtIekaNEkg IAH nig IKC pMKabegIt)anCaRtIekaNEkg EdlmanIA nig IC CaRCugmMuEkg nig KCAH CaGIb:UetnUs .

I

A CH , K

r

FrNImaRtviPaKkgbg;


tamTMnak;TMngmaRtkgRtIekaNEkgeK)an )2(IC

1IA

1r1

222 tamRTwsIbTsVyKuNcMNuc I eFobnwgrgVg; )C( eK)an 22 OIRIF.ICIE.IA naM[ )3(

IEOIRIA

22 nig )4(IF

OIRIC22

ykTMnak;TMng )3( nig )4( CMnYskg )2( eK)an

222

22

2 )OIR(IFIE

r1

naM[ )5(

r)OIR(IFIE 2222

22

ykTMnak;TMng )5( CMnYskg )1( eK)an 2

2

2222 R

r2)OIR(OI

b 0rR2r.OI2)OIR( 2222222 tag OId eK)an 0rR2rd2)dR( 2222222 b 0rR2Rd)Rr(2d 2242224 tag 2dt eK)an 0rR2Rt)Rr(2t 224222 DIsRKImINg;bRgYm

)R4r(r)rR2R()Rr(' 222224222 eKTajbs 22221 R4rrRrt

FrNImaRtviPaKkgbg;


b 22222 R4rrRrt eday Rd enaH 22 Rdt . ehtuenH 22222 R4rrRrdt naM[ 2222 R4rrRrdOI Bit . smal; tamsmPaB 0rR2rd2)dR( 2222222 eKGacsresr

])dR()dR[(r)dR(r2)dR( 222222222 EckGgTaMgBIrnwg 2222222 r)dR()dR(r)dR( eK)anTMnak;TMng

222 )dR(1

)dR(1

r1

.

FrNImaRtviPaKkgbg;


CMBUkTI2

lMhat;TI1 eK[ ABC CaRtIekaNmYyehIy D CaeCIgnkm

FrNImaRtviPaKkgbg;


tag )w( 1 nig )w( 2 CargVg;manGgt;piterogKa AB nig AC . eK)an )w(E 1 eRBaH BEAE ehIy )w(F 2 eRBaH CFAF tag P nig Q CacMnucRbsBVrvagbnat; )AN( CamYy )w( 1 nig )w( 2 erogKa . cMnuc N enAeRkArgVg; )w( 1 enaHeK)an )1(NA.NQND.NE eRBaH N CacMNuckNaln EF . cMeBaHrgVg; )w( 2 eKman )2(NA.NPND.NF tam )2(&)1( eKTaj)an NA.NPNA.NQ b NPNQ eday MCMB enaHeK)an PC//MN ehIyeday PCAP dUcenH NMAN .

FrNImaRtviPaKkgbg;


lMhat;TI2 eK[RtIekaN ABC mYymanRCug c,b,a nigmanpRkLa S . ]bmafaDEF CaRtIekaNcarwkkgRtIekaN ABC .

cUrRsaybBaak;fa abcS8FDEFDE

2

dMeNaHRsay

tag zAF,yCE,xBD enaH

cz0by0ax0

eK)an zcBF,ybAE,xaDC tamRTwsIbTkUsIunUskgRtIekaN BDF eK)an

Bcos)zc(x2)zc(xDF 222 eday )CAcos())CA(cos(Bcos

A

B C D

E

F

FrNImaRtviPaKkgbg;


FrNImaRtviPaKkgbg;


ksareyag 1-KNitviTafak;TI10PaK2rbs;RksYgGb;rM yuvCn nigkILa e)aHBum

Eulclide Geometry.pdf

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