Ccchuyn,bivitdiydo laisacsutmtrnInternetribinson,ctxnvdnthnhhaifiletnghpvimcchphgip(thigiantmkim)choccemituyntrngTHPTchuynL.V.C.Vkhnglinhc trctipvicctcgixinphp,mongthngcm!

1.nhLCevaChotamgicABC.D,E,FlnltnmtrncccnhBC,AC,AB.Chngminhrngccmnh

saultngng:1.1 AD,BE,CFngquyti mtim.

1.2

sin sin sin. . 1

sin sin sin

ABE BCF CAD

DAB EBC FCA = .

1.3 . . 1AE CD BFEC DB FA

= .

Chngminh:Chngtaschngminhrng1.1dnn1.2,1.2dn

n1.3,v1.3dnn1.1.Gis1.1ng.GiPlgiaoimcaAD,BE,CF.

TheonhlhmssintrongtamgicAPD,ta

c:

sin sin.

sin sin

ABE ABP APBPDAB BAP

= = (1)

Tngt,tacngc:

sin

sin

BCF BPCPEBC

= (2)

sin.

sin

CAD CPAPFCA

= (3)

Nhntngvca(1),(2),(3)tac1.2.Gis1.2ng.TheonhlhmssintrongtamgicABDvtamgicACDta

c:

sin sin .

sin sin

ADB AB CAD CDDB CABAD ACD

= = Do:

sin. .

sin

CAD AB CDCA DBBAD

= ( )0180BDA ADC + = (4)

Tngt,tacngc:

sin. .

sin

BCF CA BFBC FAFCA

= (5)

www.laisac.page.tl

MMTTSSCCHHUUYYNN,,BBIIVVIITTVVHHNNHHHHCCPPHHNNGG((TTppII::VVNNDDNNGGTTNNHHCCHHTT,,NNHHLLNNIITTIINNGG))

sin. .

sin

ABE BC AEAB ECEBC

= (6)

Nhntngvca(4),(5), (6)tac1.3.Gis1.3ng,tagi 1, .P CF BE D AP BC = = I ITheo1.1v1.2,tac:

1

1

. . . . 1CDAE BF AE CD BF

EC D B FA EC DB FA = = hay: 1

1

.CD CDD B DB

= Do: 1D D .

Nhnxt.VinhlCeva,tacthchngminhcccngtrungtuyn,ngcao,ngphngic

trongcatamgicngquytimtim.Ccimlnltltrngtm(G),trctm(H),tmngtrnnitiptamgic(I).NungtrnnitiptamgicABCctAB,BC,CAlnltltiF,D,E.Khi, tac:AE=AF BF=BDCD=CF.BngnhlCeva,tachngminhcAD,BE,CFngquytimtim,imgilimGergonne(Ge) catamgicABC(hnhdi).

Lu:nhlCevacthcsuyrngbinhnggiaoimnmngoitam gicABC mkhngnhtthitphinmtrongn.Vvy,ccimD,E,FcthnmngoicccnhBC,CA,ABnhhnhbn.

VdsauschothyrtcdngcanhlCeva.Biton. [IMO2001ShortList]ChoimA1ltmcahnhvungni tiptamgicnhnABCc

hainhnmtrncnhBC.CcimB1,C1cnglnltltmcacchnhvungnitiptamgicABCvimtcnhnmtrn ACvAB.ChngminhrngAA1,BB1,CC1ngquy.

Ligii:GiA2lgiaoimcaAA1vBC.B2vC2cxc

nhtngt.Theonhlhms sin,ta c:

1 1

1 1

sin

sin

SA SAAAA A SA

= hay

( )1 2

01

sin

sin 45

SA BAAAA B

= +

Tngt:

( )1 2

01

sin

sin 45

TA CAAAA C

= +

hay

( )

0

1

1 2

sin 45

sin

CAATA CAA

+ = .Do, tac:

( ) ( )

0

2 1 1

01 12

sin 45sin. . 1.

sin sin 45

CBAA AA SATA AACAA B

+ = =

+(1)

Chngminhhontontngt, ta cngc:

( ) ( )

0

2

02

sin 45sin. 1.

sin sin 45

BBCC

ACA A

+ =

+(2)

( ) ( )

0

2

02

sin 45sin. 1.

sin sin 45

AABB

CBB C

+ =

+(3)

Nhntngvca(1), (2), (3) kthpnhlCevataciucnchngminh.Bitp pdng:

1. QuaccimAvDnmtrnngtrn kccngtiptuyn,chngctnhautiimS.TrncungADly ccimA vC. CcngthngACvBDctnhautiimP, ccngthngABvCDctnhautiimO.ChngminhrngngthngPQchaimO.

2. Trn cccnhcatamgicABCvpha ngoitadngcchinhvung.A1,B1,C1, ltrungim cccnh ca cc hnhvungnminhauvi cccnhBC,CA,ABtngng.ChngminhrngccngthngAA1,BB1,CC1ngquy.

3. Chngminhccngcao,ngtrungtuyn,tmngtrnnitip,ngoitiptamgicngquy timtim.

4. Trn cccnhBC,CA,ABcatamgicABClyccimA1,B1,C1saochoccngthngAA1,BB1,CC1ngquy timtim.ChngminhrngccngthngAA2,BB2,CC2ixngviccngthng quaccngphngictngng,cngngquy.

2.nhLMenelausChotamgicABC.Ccim H,F,GlnltnmtrnAB,BC,CA.Khi:

M,N, Pthnghngkhi vchkhi . . 1.AH BF CGHB FC GA

= -

Chngminh: Phnthun:Sdngnhlsintrongcctam gicAGH,BFH,CGF, tac:

sin sin sin

sin sin sin

AH AGH BF BHF CG GFCGA HB FCAHG HFB CGF

= = = .

(vilu rng sin sin sin sin sin sin .AGH CGF AHG BHF HFB GFC = = = )Nhntngvtaciuphichngminh.

Phno: Gi ' .F GH BC = I Hontontngtta cc:

( )'. . . . 1 .'

AH BF CG AH BF CGHB F C GA HB FC GA

= = - Hay'

'BF BFF C FC

= ,suyra '.F F

Nhnxt.nhlMenelausc rtnhiu ngdngtronggiiton.Nhiunhlnitingcchngminhmt cchddngnhnhlMenelausnhnhlCeva,Pascal,Desargues(scnuphnbitpdiy).

Vd:ChoA,B,C,D, E, F lccimnmtrnmtng

trn (c thkoxptheoth tnhtrn). Gi, , .P AB DE Q BC EF R CD FA = = = I I I Chng

minhrngP,Q, Rthnghng.Chngminh:Gi , , .X EF AB Y AB CD Z CD EF = = = I I IpdngnhlMenelauschoBc,DE,FA (ivi

tamgicXYZ), ta c:

( ). . . . . . 1 .ZQ XB YC XP YD ZE YR ZF XAQX BY CZ PY DZ EX RZ FX AY

= = = -

Do: . . 1ZQ XP YRQX PY RZ

= - .TheonhlMenelausta

cP, Q, Rthnghng.

Bitp pdng:1. imPnmtrnngtrnngoitipcatamgicABC,A1,B1,C1lnltlchn

ngvunggcht PxungBC,CA,AB.ChngminhrngA1,B1,C1thnghng.2. TrongtamgicvungABCk ngcaoCKtnh ca gcvungC,cntrongtam

gicACKkngphngicCE.D ltrungim caonAC,F lgiaoimcaccngthngDEvCK.ChngminhBF//CE.

3. CcngthngAA1,BB1,CC1ngquy tiimO.Chngminhrnggiaoim caccngthngABv A1B1,BC vB1C1,CA vC1A1nmtrnmtngthng.

4. ChohaitamgicABC,ABC.NuccngthngAA,BB,CCngquy timtimO,thccimP,Q,Rthnghng,trong

' ', ' ', ' '.P BC B C Q CA C A R AB A B = = = I I I

P

Y

ZR

Q

O

FA

C

DE

B

X

DINTCHTAMGICTHYTCIvanBorsenco

Translator:DuyCuong.

TrongTonhc,HnhHclungiiquytccbitonvinhngktquchnhxcvhpdn.BivitsauytrnhbyvmttrongnhngktqutuytptrongbitonhnhhcnhlEulerchotamgicthytcvngdngcan.Chngtahybtuviphpchngminhnhl,saucngnhautholunccBitonOlympiad.

nhl1.ChoC(O,R) lngtrnngoi tiptamgicABC.XtmtimMtunmtrongtamgic.KhiuA1,B1, C1 lhnhchiucaMlnccmtcatamgicth

1 1 1

2 2

2.

4A B C

ABC

R OMS

S R

- =

Chngminh. utintarngAB1MC1,BC1MA1,CA1MB1 lnhngtgicnitip.pdngnh l Cauchy m rng vo tam gic AB1C1 ta c 1 1 sinB C AM a = . Tng t, ta c

1 1 sinAC BM b = v 1 1 sinB C CM g = .Tsuyra:

1 1 1 1 1 1, , .2 2 2

B C AM AC BM A B CMBC R AC R BC R

= = =

GisAM, BM, CMctngtrn C(O,R) ti X, Y,Z.

Tac:

1 1 1 1 1 1 1 1 1 .A B C A B M MB C ACM MAC ZYB BYX ZYX = + = + = + = Tng t,

1 1 1B C A YZX = v 1 1 1 .B AC YXZ = Do, 1 1 1A B C D ngdngvi XYZ D v 1 1 11 1 .A B CRA B

XY R =

Mtkhc, MAB D ngdngvi MYX D nntac .XY MXAB MB

=

Tccktquthuvtac:

1 1 1

1 1 1

2 2

1 1 1 1 1 12 2

. . .. . . .

. . 2 2 4 4A B C

ABC A B C

R OMS R A B B C AC MX MA MB MA MXS R AB BC AC MB R R R R

- = = = =

Ttacththybiuthctrnkhngphthucvovtrim M(kctronghayngoingtrn).

Hqu1:Nu M nmtrnngtrn,hnhchiucaMlncccnhtamgicsthnghng(nhlSimson).

Mtnhlnamchngtimungiithiu(khngchngminh)nbnclnhlLagrangeniting.

nhl2 ChoMlmtimnmtrongmtphngtamgicABC vibs ( , , )u v w .Vimtim P btknmtrongmp(ABC).Tac:

2 2 22 2 2 2. . . ( ) .

vwa uwb uvcu PA v PB w PC u v w PM

u v w + +

+ + = + + + + +

CchchngminhcthtmthykhipdngnhlStewartmtsln.nhlnycmthqurthaykhi P trngvitm OngtrnngoitiptamgicABC.Khi,tac:

2 2 22 2

2.

( )vwa uwb uvc

R OMu v w + +

- = + +

ThqunykthpnhlEulerchotamgicthytc,tacnhlsau:nhl3:ChoMlimnmtrongmtphngtamgicABC,vibs(u,v,w).KhiuA1,B1,C1

lhnhchiucaMlncccnhtamgic.Tac:

1 1 1

2 2 2

2 2.

4 ( )A B C

ABC

S vwa uwb uvcS R u v w

+ + =

+ +Tktqutrnchngtacththynhl1vnhl3achngtanhnthcphnnov

dintchtamgicthytc.nhlEulervdintchtamgicthytcthtslmtcngctinchcho vic gii cc bi ton hnh hc.ng dng u tinm chng ti sp gii thiu sau y ni vnhngimBrocard

nhnghaimBrocard:TrongtamgicABC,ngtrnquaAvtipxcvi BCti B,ngtrnquaBvtipxcviACtiCvngtrnquaCtipxcviABtiA.Chngctnhautimtim,gi limBrocard.Mtcchtngqut,tachaiimBrocard,imthhaixuthinkhitaquayngcchiukimnghstipxccaccngtrn.

Biton1:Cho 1 W v 2 W lhaiimBrocardcatamgicABC.Chngtrng 1 2O O W = W ,viOltmngtrnngoitiptamgicABC.

Hngdn: Tnhnghaim Brocard,tathy:

1 1 1 1AB BC CA w W = W = W = ,tngt 2 2 1 2BA AC CB w W = W = W = .Taichngminh 1 2w w = .

rng 12

1 1 12

. sin sin. sin sin

B C

ABC

S B BC w wS AB BC b b

W W = = vtngt,

1 1

2 21 1

2 2

sin sin, .

sin sinC A A B

ABC ABC

S Sw wS S g a

W W = =

Cngccdintchtac:

1 1 1

2 2 21 1 1

2 2 2

sin sin sin1

sin sin sinB C C A A B

ABC

S S S w w wS a b g

W W W + + = = + + hay

2 2 2 21

1 1 1 1sin sin sin sinw a b g

= + + (1)

Cchtnhtngt,tacngc

2 2 2 22

1 1 1 1sin sin sin sinw a b g

= + + (2)

T(1),(2)suyra 1 2 .w w w = =tngchocchchngminh trn bt ngun tnh lEulercho tamgic thy tc. rng

1 W v 2 W lunnmtrongtamgicABC,bivminangtrnnhtrnuthucmp(ABC).T, 1 W v 2 W lunnmtrongtamgic.chngminh 1 2O O W = W tachngminhdintchtamgic

thytccachngbngnhau.Nuvy,tac 2 2 2 21 2R O R O - W = - W .Tsuyrapcm.Khiu 1 1 1, ,A B C lnltlhnhchiuca 1 W lncccnhBC,CA,AB.Sau,sdngnhl

hm Sin m rng, ta c 1 1 1sinAC B b = W , v 1BW l ng knh ng trn ngoi tip t gic

1 1 1BA C W .CngpdngnhlSinvotamgic 1ABW tac:

1

sin sinB c

w b W

=

Dnn: 1 1 1sin sin .AC B b c w = W =Tngt,tac 1 1 sinB C b w = v 1 1 sinA B a w = .Ddngtathytamgic 1 1 1A B Cngdngvi

tamgicABC theotsngdng sinw .T 1 2w w w = = , ta kt lun c tam gic thy tc cha 1 W v 2 W c cng din tch. Suy ra

1 2O O W = W .Ch:Giaocaccngitrungtrongtamgickhiul KvcgilimLemoine.Ta

c th chngminh c 1 2K K W = W . Hn th na, cc imO, 1 W , K, 2 W nm trn ng trnngknhOKgilngtrnBrocard.ChngtahycngnhautmhiumtBitonthvkhcsauy:

Biton2:ChotamgicABC,khiuO,I,Hlnlt ltmngtrnngoitip,nitipvtrctmcatamgicABC.Chngminhrng:

.OI OH

Hngdn:NhiucchchngminhsbtutOIhayOH,nhngchngtahychnmthngkhc,ltdintchnitipvoIvH.Mtcchngin,tmngtrnnitiplunnmbntrongtamgicABC,vnu OH R ,taciucnchngminh.Vvy,gis OH R

2 2

2

2 1.

9 4 4R OG

R -

Btngthcbnphilunng.Taichngminhbtngthcbntring.Nhlibiuthcmchngtabit

2 2 29 (1 8cos cos cos )OG OH R a b g = = -VtamgicABCnhnnntaccos cos cos 0 a b g ,dnn8cos cos cos 0 a b g .

Vvy, 2 29OG R hay2

2 .9R

OG

anktlunsau:Biton4.(Tonixng,IvanBorsenco)Viim M btknmtrongtamgicABC,xc

nhbba( 1 2 3, ,d d d )lnltlkhongccht M ncnh , ,BC AC AB .Chngminhrngtphp

ccim M thamniukin 31 2 3. .d d d r ,vi r lbnknhng trnni tip tamgic, nmtrongvngtrntmO bnknhOI.

Ligii.Gi 1 1 1, ,A B C lhnhchiuca M trncccnh , ,BC AC AB .Xt 1 1 1A B CV ltamgicthytccho

imM ,tac 1 1 1 1 1 1180 , 180 , 180B MC A MC A MB a b g = - = - = - ,nn:

1 1 1 1 2 3 1 3 1 22 2 . .sin . .sin . .sinA B CS S d d d d d d a b g = = + + V

Vitgnli,tac:

1 1 1

1 2 31

1 2 3

. .2 2 . .

2A B Cd d d a b c

S SR d d d

= = + +

V

Mtabit: 1 2 32 2 . . . .ABCS S a d b d c d = = + + VpdngBTCauchySchwarztac:

( ) ( )2

1 2 31 2 31 1 2 3

1 2 3

. . .. .4 . . . . .

2 2

d d d a b cd d d a b cS S a d b d c d

R d d d R + +

= + + + +

SdngnhlEulerchotamgicthytc,tac:

( ) ( ) ( )2 22 2 2 31 2 32

4 . . . ..

4 2 2

S R OM d d d a b c r a b c

R R R

- + + + +

Tycthsuyra 2 2 22 .OI R Rr OM = - Vyvimi imM trongtphp,tacOM OI (iuphichngminh).Biton5:(IvanBorsencongh)Cho M limnmtrongtamgic ABC vctal ( ) x y z .Gi MR lbnknhngtrn

nitiptamgicthytccaimM .Chngminhrng:

( )2 2 2

. 6 3. .Ma b c

x y z Rx y z

+ + + +

Ligii.Khiu N lnggiccaimM .Chngtacn2bsau:

B1:Nu 1 1 1A B C v 2 2 2A B C l2tamgicthytcca2nggicimM v N th6imnynmtrnmtvngtrn.

Chngminh.Chngtaschngminhrng

1 2 1 2B B C C lmttgicnitip.t BAM CAN f = = .Vbiv 1 1AB MC nitip,nn:

1 1 1 1 2 190 90 90 .

o o oAB C C B C C AM f = - = - = -

Tngt,v 2 2AB NC nntac:

2 2 1 2 2 290 90 90o o oAC B B C B B AN f = - = - = -

Do 1 1 2 2AB C AC B = nn 1 2 2 1B B C C lmttgicnitip.Cthtathucrng 1 2 2 1A A B B v

1 2 2 1A A C C cngltgicnitip.Xt3ngtrnngoi tiptgiccachngta,nuchngkhngtrngnhauthchngcmttmngphng,limgiaonhauca3trcngphng.Tuynhin,chngtacththyrngnhngtrcngphngny,tcccngthng 1 2 1 2 1 2, ,A A B B C C ,tothnh

mt tamgic,t tn l ABC ,mtiu tri ngc.iunychng t,ccim 1 2 1 2 1, 2, , , ,A A B B C C

nmtrncng1ngtrn.B2: Nu ,M N lhainggicim,thtalunc:

. . .1.

AM AN BM BN CM CNbc ac ab

+ + =

Chngmin.Gi 1 1 1A B C ltangicthytccaimM .Ddngchngminhcrng

1 1 .B C AN ^ Vydintchcatgic 1 1AB NC ctnhtheocngthc 1 11. .

2B C AN.V

1 1 .sinB C AM a = ,chngtac 1 11. . .sin

2AB NCS AM AN a = .Tngt,tatm crng

1 1

1. . .sin

2BC NAS BM BN b = v

1 1

1. . .sin

2CA NBS CM CN g =

Ttrn,tasuyra:1 1 1 1 1 1ABC AB NC BC NA CA NB

S S S S = + +

Hay:1.( . .sin . .sin . .sin )

2ABCS AM AN BM BN CM CN a b g = + +

SdngnhlhmSin,taciucnchngminh:. . .

1.AM AN BM BN CM CN

bc ac ab + + =

QuaytrliBiton.pdngBTAMGMvob2,tac:3

2 2 2

1 1 . . . . . . . .

27 27

AM AN BM BN CM CN AM AN BM BN CM CN

bc ac ab a b c = + +

Lic:

1 1 1 1 1 1 2 2 2 2 2 21 1 1, , , , ,sin sin sin sin sin sin

B C AC A B B C A C A BAM BM CM AN BN CN

a b g a b g = = = = = =

Tathuc:

1 1 1 1 1 1 2 2 2 22 2 2 2 2 2

. . . .127 .sin .sin .sin

A B B C AC B C A Ca b c a b g

V4

2 2 2 2 2 22

4.sin .sin .sin

Sa b c

R a b g = ,nn:

4

1 1 1 1 1 1 2 2 2 2 2 22

1 4. . . . . .

27S

A B B C AC A B B C A CR

Sdngb1vnhlEulerchodintchcamttamgicthytc,tac:2 2

1 1 1 1 1 1 2. . 4 . .

4MR OM

A B B C AC R SR

- =

2 2

2 2 2 2 2 2 2. . 4 . .

4MR ON

A B B C A C R SR

- =

Vy:

( ) ( )2 2 2 2 22 2

127 4 .

MR R OM R ON

R S

- -

Bctiptheolsdngnhl3choimM vnggiclinhpcan:

( ) ( )

2 2 2

2 2 22 2

2 2

a b cxyz

x y zyza xzb xycR OM

x y z x y z

+ + + + - = =

+ + + +

nggicimca M , N c2 2 2a b cx y z

+ +

ltakhnggian,vy:

( ) ( )

( )2 2 2 2 2 22 22 2 2 22 2 2

. . ..

.

a b c xyz x y z a b c x y zR ON

a b cyza xzb xyc xyzx y z

+ + + + - = =

+ + + +

Tnghp2ktqutrnvBTchngminh,tac:

( )

2 2 2 2

2 2 22 2

.1.

274 . . .

MR a b c

a b cR S x y z

x y z

+ + + +

Cuicng,tathuc:

( )2 2 2

. 6 3. .Ma b c

x y z Rx y z

+ + + +

NHLPTOLEMYDatheobicaZaizaiTrndindntonhc

1.Mu.Hnhhclmttrongnhnglnhvctonhcmanglichongiyutonnhiuiuthvnht

vkhkhnnht.Nihi taphicnhngsuyngh sngtovtinht.Trong lnhvcnycngxuthinkhngtnhngnhl,phngphpnhmnngcaotnhhiuqutrongqutrnhgiiquytccbiton,giptachinhphcnhngnhningghvhimtr.TrongbivitnyzaizaixingiithiunccbnmtviiucbnnhtvnhlPtolemytrongvicchngminhccctnhcahnhhcphng.

2.Nidung Lthuyt:2.1.ngthcPtolemy:Chotgic ABCD nitipngtrn ( )O .Khi . . .AC BD AB CD AD BC = +

Chngminh.

LyM thucngchoACsaocho .ABD MBC =Khixt ABD D v MBC D c: ,ABD MBC = .ABD MCB =Nn ABD D ngdngvi ( . ).MBC g g DDotac:

. .AD MC

AD BC BD MCBD BC

= = (1).LicBA BMBD BC

= v ABM DBC =

Nn ( . ).ABM DBC g g D D : Suyra AB BDAM CD

= hay . .AB CD AM BD = (2).

T(1)v(2)suyra . . . . . .AD BC AB CD BD MC AM BD AC BD + = + =VyngthcPtolemycchngminh.2.2.BtngthcPtolemy.ycthcoilnhlPtolemymrngbivnkhnggiihntronglptgicnitip.

nhl:Chotgic ABCD .Khi . . . .AC BD AB CD AD BC +Chngminh.

TrongABC lyimMsaocho: ,ABD MBC = ADB MCB =

Ddngchngminh:

. .

AD BDBAD BMC

MC CBBD CM AD CB

D D =

=

:

Cngtktluntrnsuyra:

,AB BDBM BC

= ABM DBC =

( . . )

. . .

ABM DBC c g c

AB BDAB DC BD AM

AM CD

D D

= =

:

pdngbtngthctrongtamgicvcciutrntac:

. . ( )AD BC AD DC BD AM CM DBD AC + = + >VynhlPtolemymrngcchngminh.3,ngdngcanhlPtolemy.Muchophnnychngtasnvi1vdinhnhvcbnvvicngdngnhl

Ptolemy.Biton1. [thivotrngTHPTchuynLQun,thxngH,tnhQungTr,nmhc

20052006]ChotamgicuABC ccccnhbng ( )0 .a a > Trn AClyimQ ding,trntiaicatiaCB lyim P dingsaocho 2. .AQ BP a = GiM lgiaoimca BQ v AP .Chngminhrng .AM MC BM + =

Chngminh.Tgithit 2.AQ BP a = suyra

.AQ ABAB BP

=

Xt ABQ D v BPA D c:

( )AQ AB

gtAB BP

= v BAQ ABP =

Suyra ( . . )ABQ BPA c g c D D : ( )1ABQ APB =

Lic 060 (2)ABQ MBP + =T (1), (2) tasuyrac 0 0 0 0 0 0180 120 180 180 120 60 .BMP MBP MPB AMB BMP ACB = - - = = - = - = =Suyratgic AMCB nitipcngtrn.pdngnhlPtolemychotgic AMCB nitipvgithit .AB BC CA = =Tac . . .AB MC BC AM BM AC AM MC BM + = + = (pcm).

yl1bitonkhdvttnhincchgiinykhngcnginlm.VnumunsdngngthcPtolemytrong1kthithclphichngminhndidngb.Nhngiuchyltachngcnphisuynghnhiukhidngcchtrntrongkhinudngcchkhcthligiickhilikhngmangvtngminh.

Bi ton 2. [ thi chn i tuyn Hng Kng tham d IMO 2000] Tam gicABC vungcBC CA AB > > .GiD lmtimtrncnh ,BC E lmtimtrncnh AB kodivphaimA saocho .BD BE CA = = .GiP lmtimtrncnh ACsaocho , , ,E B D P nmtrnmtngtrn.Q lgiaoimthhaica BP vingtrnngoitip ABC D .Chngminhrng: AQ CQ BP + =

Chngminh.Xtcctgicnitip ABCQ v BEPD tac:

CAQ CBQ DEP = =(cngchncccungtrn)

Mtkhc 0108AQC ABC EPD = - =Xt AQC D v EPD D c:

,AQC EPD =

. . . (1)

CAQ DEP AQC EPD

AQ CAAQ ED EP CA EP BD

EP ED

= D D

= = =

:

(do AC BD = )

. . . (2)AC QC

ED QC AC PD BE PDED PD

= = =

(do AC BE = )pdngnhlPtolemychotgicnitip BEPD tac:. . . (3)EP BD BE PD ED BP + =T(1),(2),(3)suyra . . .AQ ED QC ED ED BP AQ QC BP + = + = (pcm).Cththyrngbi1lttngngintaxydngcchgiicabi2.Tcldavocc

ilngtrongtamgicbngnhautheogithittasdngtamgicngdngsuyracctslinquanvsdngphpthsuyraiuphichngminh.Cchlmnytrakhlhiuquvminhharrngqua2vdmzaizainutrn.lmrhnphngphpchngtascngnhaunvivicchngminh1nhlbngchnhPtolemy.

Biton3.(nhlCarnot)Chotamgicnhn ABC nitiptrongngtrn ( , )O R vngoitipngtrn ( , ).I r Gi , ,x y z lnltlkhongcchtO ticccnhtamgic.Chngminhrng:x y z R r + + = +

Chngminh.Gi , ,M N P lnltltrungimca , , .BC CA AB Gis

,x OM = ,y ON = ,z OP = ,BC a = ,CA b = .AB c =TgicOMBP nitip,theongthcPtolemytac:

. . .OB PM OP MB OM PB = +

Do: . . . (1)2 2 2b a c

R z x = +

Tngttacngc:

. . . (2)2 2 2

. . . (3)2 2 2

c a bR y x

a c bR y z

= +

= +

Mtkhc:

( )

2 2 2

. . . 42 2 2

ABC OBC OCA OAB

a b cr S S S S

a b cx y z

+ + = = + + =

= + +

T(1),(2),(3),(4)tac:

( ) ( )2 2

a b c a b cR r x y z

R r x y z

+ + + + + = + +

+ = + +

ylmtnhlkhquenthucvcchchngminhkhngin.ngdngcanhlnynhnildngnhiutrongtnhtonccilngtrongtamgic.ivitrnghptamgickhngnhnthcchphtbiucanhlcngcsthayi.

Biton4. [ThiHSGccvngcaM,nm1987]Chomttgicnitipccccnhlintipbng , , ,a b c dvccngchobng , .p q Chngminhrng

2 2 2 2( )( ).pq a b c d + +Ligii.pdngnhlPtolemychotgicnitipth ac bd pq + =Vytacnchngminh 2 2 2 2 2 2 2( ) ( )( )p q ac bd a b c d = + + +BtngthcnychnhlmtbtngthcrtquenthucmclaicngbitlBT

CauchySchwarz.Vybitoncchngminh.Mtligiipvvcnggnnhcho1bitontngchngnhlkh.tngyla

btngthccnchngminhv1dngnginhnvthunishn.ThtthvlbtngthclilBTCauchySchwarz.Biton5.Chongtrn ( )O vBClmtdycungkhcngknhcangtrn.Tm im

AthuccunglnBCsaocho AB AC + lnnht.Ligii.GiDlimchnhgiacungnhBC.t DB DC a = = khngi.TheonhlPtolemytac:

. . . ( ) .BC

AD BC AB DC AC BD a AB AC AB AC ADa

= + = + + =

DoBCvAkhnginn AB AC + lnnhtkhivchkhi AD lnnhtkhivchkhiAlimixngcaDquatmOcangtrn.

4.Bitp.

Biton 4.1. [CMO1988,TrungQuc]Cho ABCD lmttgicnitipvingtrnngoitipctm(O)vbnknhR.Cctia , , ,AB BC CD DA ct ( , 2 )O R lnltti ', ', ', '.A B C D Chngminhrng:

' ' ' ' ' ' ' ' 2( )A B B C C D D A AB BC CD DA + + + + + +Biton4.2.Chongtrn ( )O vdycungBCkhcngknh.Tm imAthuccungln

caBCngtrn 2AB AC + tgitrlnnht.Biton4.3. ChotamgicABCnitipngtrn ( ).O ngtrn ( ')O nmtrong ( )O tipxc

vi ( )O tiTthuccungAC(khngchaB).Kcctiptuyn ', ', 'AA BB CC ti ( ').O Chngminhrng'. '. '.BB AC AA BC CC AB = +Biton4.4. CholcgicABCDEFccccnhcdinhhn1.Chngminhrngtrongba

ngcho , ,AD BE CFctnhtmtngchocdinhhn2.Bi ton4.5. Chohaingtrn ngtm,bnknhcangtrnnygpibnknhca

ngtrnkia. ABCD ltginitipngtrnnh.Cctia , , ,AB BC CD DA lnltctngtrnlnti ', ', ', '.A B C DChngminhrng:Chuvitgic ' ' ' 'A B C D lnhn2lnchuvitgic .ABCDNHLPTOLEMYMRNG

1. nhlPtolemymrng.Cho ABC D ni tipngtrn ( ).O ngtrn ( )1O thayilun tipxcvi BC(khngcha A ).Gi ', ', 'AA BB CC ln lt lcc tip tuyn t , ,A B C nngtrn ( )1 ,O thtachhthcsau

. ' . ' . '.BC AA CA BB AB CC = +Chngminh.

Xt trng hp ( )O v ( )'O tip xc ngoivi nhau (trng hp tip xc trong chngminhtngt).

GisMltipimca ( )O v ( )' .O MA,MB,MCtheoth tct ( )'O tiX,Y,Z.

Lc / / , / / , / / .YZ BC XZ AC XY AB TheonhlThalstac

( )1AX BY CZAM BM CM

= =

Lic ( )2 2 2' . , ' . , ' . 2AA AM AX BB BM BY CC CM CZ = = =

T(1)v(2)suyra2 2 2

2 2 2

' ' 'AA BB CCAM BM CM

= = hay' ' 'AA BB CC

AM BM CM = =

TnhlPtolemychot gicnitip MCAB thuc . . .BC MA CA MB AB MC = +Do . ' . ' . '.BC AA CA BB AB CC = +2. Ccbitonngdng.

AC

BZ

YX

OO M

B

A

C

Biton2.1.Cho2ngtrntipxcnhau,ngtrn lnvtamgacuni tip.Tccnhcatamgackcctiptuyntingtrnnh.Chngminhrngdimttrongbatiptuynbngtnghaitiptuyncnli.

Ligii.

Coi rng ng trn nh tipxcvi BC (khng chaA) vt , ,a b cl l l ln lt l di ccngtiptuynkt , ,A B C nngtrnnh.

TnhlPtolemymrngtac.a b c a b cal bl cl l l l = + = +

Biton2.2.ChohnhvungABCDni tipngtrn ( )O .Mtngtrnthayi tipxcvionCDvcungnhCD,ktiptuynAX,BYvingtrnny.ChngminhrngAX+BYkhngi.

Ligii.Khngmttnhtngqut,coirnghnhvungABCD

c di cnh bng 1. T d dng suy ra

2.AC BD = =p dnh nh l vi cc tam gac ACD v BCD, ta

c

. . .

. . .

CD AX AD CM AC DM

CD BY BC DM BD CM = +

= + . Cnghaingthc

ny,suyra

1 2.AX BY + = +Biton2.3. ( )ABC AB BC D

a b cal bl cl = + hay

( ) ( ) ( )2 .cos 2 .cosx x c A b c x c b x c A + = - + - -Suyra ( ) ( )22 .cos .cos .a b c x bc c A ac A + + = - -Gi y lditiptipt 1B tingtrnnitip 1 .BB C D

Ththtac ( ) ( ) ( )1 11 1 2 .cos2 2y BB B C BC c b c A a = + - = + - -Lic

( ) ( ) ( ) ( )

2

2 2 2 2 2

2 2 2 .cos .cos 2 .cos

2 2 .cos 2 .cos .

R r x y bc c A ac A a b c c b c A a

bc b c a bc A a b c bc A

= = - - = + + + - -

= + - - = + -

Tytaciuphichngminh.3. Bitp.

3.1. [TH&TT bi T8/369] Cho ABC D ni tip ng trn ( )O v c di cc cnh, , .BC a CA b AB c = = = Gi 1 1 1, ,A B C theothtlimchnhgiacung BC (khngchaA), CA

(khngchaB), AB (khngchaC).Vccngtrn ( ) ( ) ( )1 2 3, ,O O O theothtcngknhl1 2 1 2 1 2, , .A A B B C C Chngminhbtngthc

( ) ( ) ( ) ( )

1 2 3

2

/ / / .3A O B O C Oa b c

P P P + +

+ +

ngthcxyrakhino?

3.2. [TH&TTbiT11/359]Cho .ABC D ngtrn ( )1O nmtrongtamgicvtipxcvicccnh , .AB AC ng trn ( )2O iqua ,B C v tipxc ngoi ving trn ( )1O ti .T Chngminhrngngphngiccagc BTC iquatmngtrnnitipca .ABC D

MTTSVNG DNGDatheobicaSonHongTa

TrnMathematicalreflections2(2008)

TrongbinhlPtolemymrngtathymttnhchtthvivihaingtrntipxcnhau,binytatiptctmhiuthmvmttnhchtkhcivihaingtrntipxctrongvinhauvsngdngcatnhchtny.

B. Cho Av B lhaiimnmtrnngtrn. g Mtngtrn r tipxctrongvi g ti .T Gi AE

v BF lhaitiptuynkt Av Bn r .Thtac

.TA AETB BF

=

Ligii.Gi 1A v 1B lgiaoimthhaica ,TA TBvi r .Chngtabitrng 1 1A B songsongvi .AB V thtac,

F

E

B1

A1

T

A

B

2 2

1 1

1 1 1 1 1 1

.. .

.AE AA AT BB BT BFTA AT AT BT BT TB

= = =

Suyra 1

1 1 1

,AE BF AE TA TATA TB BF TB TB

= = = ychnhliuphichngminh.

minhhachobny,chngtasnvimtvivd.BitonsauycnghcaNguynMinhH,trongtpchtonhcvtuitr(2007).

Biton1.Cho W lngtrnngoitipcatamgic ABC v D ltipimcangtrn ( )I r nitiptamgic ABC vicnh .BC Chngminh

rng 090 .ATI =Ligii.GiE v F lnltlhaitipimca

ngtrn ( )I r vicnhCAv .AB Theobtrntac,

.TB BD BFTC CD CE

= =

VvytamgicTBF vTCE ngdng.Suyra,TFA TEA = cnghal , , , ,A I E F T cngnmtrn

mtngtrn.Vy 090 .ATI AFI = =

Biton2.Cho ABCD ltgicnitiptrongngtrn . W Cho w lngtrntipxctrongvi W ti ,Tvtipxcvi ,BD AC ti , .E F Gi P lgiaoimca EF vi .AB ChngminhrngTP lngphngictrongcagc .ATB

Ligii. Tb,chngtacc ,AT AFBT BE

= vvybitonscchngminhnutac

.AF APBE PB

=

A

T

CB

IF

D

E

EP

A

C

D

B

T

F

Chrng ,PEB AFP = vtnhlhmsinivihaitamgic , ,APF BPE chngtac

csin sin

.sin sin

AP AFP BEP BPAF APF BPE BE

= = =

Vvy,tac ,AF APBE PB

= suyraiuphichngminh.

Sau y l ton t Moldovan Team Selection nm2007.

Bi ton 3. Cho tam gic ABC v W l ng trnngoitiptamgic.ngtrn w tipxctrongvi W ti,T vvi cnh ,AB AC , .P Q Gi S l giao im ca

AT vi .PQ Chngminhrng .SBA SCA = Ligii.Dngb,tac

sin sin.

sin sinBP BT BCT BAT PSCQ CT CBT CAT QS

= = = =

Tyddngsuyrahaitamgic BPSv CQS ng

dng.Vvy .SBA SCA =

Biton4.Chongtrn ( )O cdycung AB .Gi ( ) ( )1 2,O O lhaingtrntipxctrongvi ( )O v AB .Gigiaoimgia ( ) ( )1 2,O O l , .M N Chngminhrng MN iquatrungimcung AB (khngcha ,M N ).

Ligii.Gi PvQ lnltltipimcang trn ( )1O vi ( )O v .AB R vS lnlcltipimcangtrn ( )2Ovi ( )O v .AB Cho T l trung im cacung AB (khngcha ,M N ).

p dng b i vi hai ng trn ( ) ( )1,O O vhaiim ,A B vihaitiptuyn

,AQ BQ tingtrn ( )1O ,thchngtac

.PA QAPB QB

=

iunycnghal PQ iqua .TTngt, RS cngiqua .TMtkhc,talic

,

PQA QTA QAT

PRA ART PRS = + = = + =

P

N

M

T

BA

SQ

R

S

B

P

C

A

T

Q

K

M

A

QP

T

BC

Nnbnim , , ,P Q R S cngnmtrnmtngtrnvginlngtrn ( )3O .Ch rng PQ l trcngphng ca ( )1O v ( )3 ,O RS l trcngphngca ( )2O v

( )3 ,O cn MN ltrcngphngca ( )1O v ( )2 .O Vvybangthng , ,PQ RS MN sngquytitmngphngcabangtrnny.

Vytasuyrac MN siqua .TChngtastiptcvimtbitontrongMOSPTestsnm2007.Biton5.Chotamgic ABC .ngtrn w iqua , .B C ngtrn 1 w tipxctrongvi w

vhaicnh ,AB AC lnltti , , .T P Q Gi M ltrungimcacung BC(chaT)cangtrn. w Chng minh rng ba ng

thng , ,PQ BC MT ngquy.Ligii.Gi K PQ BC = v' .K MT BC = p dng nh

l Menelaos trong tam gicABC ,tac

. . 1

.

KB QC PAKC QA PB

KB BPKC CQ

=

=

Mt khc, M l trung imca cung BC(cha T) ca wnn MT l ng phn gicngoi ca gc .BTC V vy,bitonscchngminhnu

ta c .BP TBCQ TC

= Nhng iu

ny ng theo b , nn ta c iu phichngminh.

Biton6.Cho ( ) ( )1 2,O O lhaingtrn tipxc trongving trn ( )O ti

, .M N Tip tuyn chung trong ca haingtrnnyct ( )O bnim.Gi Bv C lhai trongbnimtrnsaocho Bv C nmcngphavi 1 2.O O Chngminhrng BC song song vi mt tip tuynchungngoicahaingtrn ( ) ( )1 2, .O O

Li gii. V hai tip tuyn chung trong,GH KL ca ( ) ( )1 2,O O saocho ,G L nm

trn ( )1O v ,K H nmtrn ( )2 .O GiEF

M

YX

AC

K

L

B

H

G

QP F

E N

ltiptuynchungngoica ( ) ( )1 2,O O saocho ,E B nmcngphasovi 1 2.O O ,P Q lgiaoimca EF vi ( ).O Giytachcnchngminh BC songsongvi .PQ Gi A ltrungimcungPQ khngcha , .M N Vhaitiptuyn ,AX AY nngtrn ( ) ( ) ( ) ( ) ( )1 2 1 2, , .O O X O Y O Trongbiton4tachngminhcrng , ,A E M thnghng , ,A F N thnghng,v MEFN ltgicnitip.Vvy,

2 2. .AX AE AM AF AN AY = = = hay .AX AY =

Theobtac, .MA MB MCAX BG CL

= = MtkhctheonhlPtolemy,tac

. . . ,MA BC MB AC MC AB = +Suyra . . . .AX BC BG AC CL AB = +Tngttacngc . . . .AY BC BH AC CK AB = +Nn ( ) ( ). . ,AC BH BG AB CL CK - = - hay . . ,AC GH AB KL = suy ra .AC AB = iu ny c

nghal A ltrungimcung BC cangtrn ( ).OVvy BC// .PQ

CCVICC

1.1. nhngha:

Chongtrn(C)tmObnknhr,mtimPngoi(C).TrnOPlyPsaochoOP.OP2=r2.

Tani:i cccaP lng thngd vunggcviOP tiP.Ngc li, viming thngd

khngquatmO,taniPlcccangthngd.

1.2. Tnhcht:

1. ViimPngoi(O,r),tPk2tiptuynPX,PYca(O,r).GiPlgiaoimcaXYv

OP.Tac OP XY ^ .KhitaniiccdcaimPlngXY.Ngcli,vihaiimphn

O PP'

X

Y

bitX,Ytrnngtrn(O,r),cccaXYlimP(imgiaonhaucahaitiptuyntiX,Yca

(O,r).imPnmtrnngtrungtrccaonthngXYv 90oOXP OYP = = .2. Chox,ylnlilicccaX,Y,tac X y Y x (nhl LaHire).

3. Chox,y,zlnltlicccabaimphnbitX,Y,Z,tac Z x y z XY =

Chng minh. S dng nh l La Hire, ta c Z x y = X thuc z v Y thuc z

z XY =

4. ChoW, X, Y, Z nm trn (O,r). i cc p ca WZP XY = l ng thng qua im

WXQ ZY = v ZR X YW = .

Chngminh: lyS,T ln lt l cccas=XY, t=WZ, P s t = .Sdng tnhcht (3. ),

, wS x y T z = = v p ST = . Vi lc gic WXXZYY, ta c:

WX , X ,Q ZY S X YY R XZ YW = = =

XXltiptuyntiX.tacS,Q,Rcngthucmtngthng.Tngtivi lcgic

XWWYZZ,tathyQ,T,Rcngthucmtngthng.Ttasuyracp=ST=QR.

Mtsvdngdngcctnhchtccicc:

1.3.1 ChonangtrntmOngknhUV.P,Qlhaiimthucnangtrn saocho

UP

lLaHire,tacRnmtrnngicccaK,doicccaKlngRSvKthucng

thngchangknhUVnntac SR UV ^ .1.3.2 ChotgicABCDvngtrn(O,r)nitiptgic.GiG,H,K,Llnltltipim

caAB,BC,CD,DAvi(O).KodiAB,CDctnhautiE,ADvBCctnhautiF,GKvHLct

nhautiP.Chngminhrng EFOP ^ .

Hnggii.

Sdngtnhcht1vccvicc,tacicccahaiimphnbitE,FlGKvHL.Li

cGKctHLtiP.pdngtnhcht3tacicccaPlEF.Theonhnghavicctac

EFOP ^ .1.3.3 ChonangtrntmOngknhAB.Clmtimnmngoingtrn.TCvct

tuynct(O) theothttiD,E.Gi(O1)ltmngtrnngoitiptamgicOBDcngknh

OF.ChngminhbnimO,A,E,Gcngthucmtngtrn.

Hnggii.

P

E

D

CB

A

H

O1

Q

O

F

O

A

B

C

DK

PH

G

L

E

F

KodiAEctBDtiP,theotnhcht4tacicccaPvingtrntmOlngthng

quaCvHviHlgiaoimcaADvEB.TacOP CH ^ .LyQlgiaoimcaOPvCH.

Li c 090PQH PDH PEH = = = , nn P, E, Q, H, D cng thuc mt ng trn, m

PQD PED DBO = = ,nnQ,D,P,Ocngnmtrnmtngtrn.TtasuyraQtrngG,v

ylgiaoimcangtrnngoitiptamgicOBDvngtrn ngknhOC.

V D . D. .P B OG PE PA P PB PG PO = = = nnO,A,E,Gcngthucmtngtrn.1.3.4 ChotgicliABCDnitipngtrntmO,ElgiaoimcaACvBD.Chnmt

imPthuc(O)saocho D B C 90oPA PC PB PDC + = + = .ChngminhrngO,P,Ethnghng.Hnggii.

Gi ( ) ( ) ( )1 2, ,O O O lnltlccngtrnngoitipABCD, , DPAC PB D D .

Ta c i cc ca 1O i vi ( )O l ngAC.

Tac:

360 ( )

270 90 D

o

o o

APC PAB PCB ABC

ABC A C

= - + + =

= - = +

V 1 2.(180 )oAO C APC = -

2.(90 D )o A C = -

180 2 Do A C = - 180o AOC = -

Nn 1 180oAOC AO C + = .Tngt,iccca 2O vi(O)lBD.Theotnhcht3tac:ElgiaoimcaACvDB,nn

icccaEivi(O)l 1 2O O .Vth 1 2.OE O O ^

1.3.5 ChoIltm ngtrnnitiptamgicABC.AB,AC,BCtipxc(I)lnlttiM,L,K.

ngthngquaBsongsongviMKctLM,LKlnlttiRvS.ChngminhtamgicRISltam

gicnhn.

Hnggii.

O1O2

E

AB

O

D

C

P

TacicccaccimB,K,L,Mlnltl

MK,BC,CA,AB.Lic 'B IB MK =

V 'B MK B thuc ng i cc ca B(nhlLaHire)

SMK R ^ icccaB lRS R,B,S thnghngvccngicccachngng

quytiB.

Lic:CcngicccaK,LctnhautiC

vL,K,SthnghngnnicccabaimL,K,

Sngquy tiC.T ta suy rai cccaS l

BCvtacngcc IR 'B A ^ ,v 0IS 180 'R AB C = - .GiTltrungimcaAC,tac:

2 ' ' ' ( ' ) ( ' )B T B C B A B K KC B M MA KC MA = + = + + + = + uuuur uuuur uuuur uuuur uuur uuuuur uuur uuur uuur

V KCuuur

khngsongsongMAuuuur

, '2 2 2

KC MA CL AL ACB T

+ + < = = nnBnmtrongng

trn ngknhAC 0 0' 90 IS 90 ISAB C R R > < D ltamgicnhn.

1.3.Mtsbitppdng:

1.4.1 (AustralianPolish98):Cho6imA,B,C,D,E,Fthucmtngtrnsaochocctip

tuyntiAvD,ngBF,CEngquy.ChngminhrngccngthngAD,BC,EFhoci

mtsongsonghocngquy.

1.4.2 ChotamgicABC.ngtrnnitip(I)tipxcviBC,CA,ABlnlttiD,E,F.Kl

mt im bt k thuc ng thng EF. BK,CK ctAC,AB ln lt ti E, F. Chngminh rng

EFtipxcvi(I).

1.4.3 ChotgicABCDngoitip(O).TipimthuccccnhAB,BC,CD,DAlnltlM,

N,P,Q.AN,APct(O)tiE,F.ChngminhrngME,QF,ACngquy.

1.4.4 ChotgicABCDni tip(O).ACctBDtiI. (AOB),(COD)ctnhautiimLkhc

O.Chngminhrng 090ILO = .1.4.5 ChotamgicABC.ngtrnngknhABctCA,CBtiP,Q.CctiptuyntiP,Q

vingtrnnyctnhautiR.Chngminhrng RC AB ^ .

BS

K

CTLA

I

B'M

R

NHLPASCALVNGDNGLcThnh

GVTHPTChuynTrnPhHiPhng

Trongbivitchuynnytimuncpnmtnhlcrtnhiungdngadng,lnhlPascalvlcgicnitipngtrn.Trongthctpdng,khithayithtccim,haylkhixtcctrnghpcbittasthucrtnhiuktqukhcnhau.

Trchttaphtbiunidungnhl:

nhlPascal:Cho cc im A,B,C,D,E,F cng thuc mt ng trn (c th hon i th t). Gi

P AB DE,Q BC EF,R CD FA = = = .

Khiccim P,Q,Rthnghng.

Chngminh:GiX EF AB,Y AB CD,Z CD EF. = = =

pdngnhlMenelauschotamgicXYZ iviccngthng BCQ,DEP,FAR,tac:

( )

( )

( )

CY BX QZ1 1

CZ BY QX

FZ AX RY1 2

FX AY RZEZ PX DY

1 3EX PY DZ

=

=

=

Mtkhc,theotnhchtphngtchcamtimivingtrntac:

( )YC.YD YB.YA,ZF.ZE ZD.ZC,XB.XA XF.XE 4 = = =Nhn(1),(2)v(3)theov,tac:

( )

QZ RY PX CY.BX.FZ.AX.EZ.DY1

QX RZ PY CZ.BY.FX.AY.EX.DZ

QZ RY PX YC.YD ZF.ZE XB.XA1 5

QX RZ PY YB.YA ZD.ZC XF.ZE

=

=

Th(4)vo(5),tacQZ RY PX

1.QX RZ PY

=

Vy P,Q,Rthnghng(theonhlMenelaus).

ngthngPQRtrncgilngthngPascal ngvibim A,B,C,D,E,F .

Bngcchhonvccim A,B,C,D,E,F tathucrtnhiuccngthngPascalkhc

nhau,cthtacti60ngthngPascal.

Z

Y X

R

Q

P A B

C

D E

F

ChnghnhnhvbnminhhatrnghpccimACEBFD.Ngoirakhichoccimcthtrngnhau(khilcgicsuybinthnhtamgic,tgic,

nggic),vd E F thcnhEFtrthnhtiptuyncangtrntiE,tacnthuthmcrtnhiuccngthngPascalkhcna.

HnhvdiyminhhatrnghpccimABCDEE,ABCCDD,AABBCC:

Tip theo ta a ra cc bi ton ng dngnhlPascal:

Biton1: (nhlNewton)MtngtrnnitiptgicABCD lnlttipxcvicccnhAB,BC,CD,DA tiE,F,G,H .

Khiccngthng AC,EG,BD,FH ngquy.

Ligii:GiO EG FH,X EH FG = = .

V D l giao im ca cc tip tuyn vi ng trn ti G,H, p dng

nh l Pascal cho cc im E,G,G,F,H,H , ta

c:EG FH O,

GG HH D,

GF HE X.

=

=

=

X

O

C

D

A

B

G

E

H

F

R

Q

Y P A B

C

D

E

R

Q

P

A D

B C

Q

R

P B C

A

P

Q R

A

B

C D

E

F

SuyraO,D,X thnghng.

pdngnhlPascalchoccimE,E,H,F,F,G, tac:

EE FF B,

EH FG X,

HF GE O.

=

=

=Suyra B,X,O thnghng.

TtacB,O,D thnghng.

Vy EG,FH,BD ngquytiO .

ChngminhtngtivingthngAC taciuphichngminh.

Biton2:Chotamgic ABC ni tiptrongmtngtrn.Gi D,E ln lt lccimchnhgiacacc

cung AB,AC P limtutrncungBC DP AB Q,PE AC R = = .

ChngminhrngngthngQR chatm I cangtrnnitiptamgicABC .

Ligii:V D,E ln lt limchnhgiacacccung AB,AC nn

CD,BE theothtlccngphngiccagc ACB,ABC .Suyra I CD EB. = pdngnhlPascalchosuimC,D,P,E,B,A, tac:

CD EB I = DP BA Q =

PE AC R. =VyQ, I,R thnghng.

Biton3: (Australia2001)ChotamgicABCnitipngtrn(O),ngcaonhA,B,Clnltct(O)tiA,B,C.Dnmtrn(O),DA' BC A",DB' CA B",DC' AB C" = = = .Chngminhrng:A,B,C,trctmHthnghng.Ligii:pdngnhlPascalchosuimA,A ',D,C ',C,B, tac:

AA ' C 'C H,

A 'D CB A",

DC ' BA C".

=

=

=

I R

Q E

D A

B C

P

H C"

B"

A"

C' B'

A'

B C

A

D

VyH,A",C" thnghng.

TngtsuyraA,B,C,Hthnghng.

Biton4: (IMOShortlist1991)PthayitrongtamgicABCcnh.GiP,PlhnhchiuvunggccaPtrnAC,BC,Q,QlhnhchiuvunggccaCtrnAP,BP,giX P 'Q" P"Q ' = .

Chngminhrng:Xdichuyntrnmtngcnh.Ligii:Tac: 0CP 'P CP"P CQ 'P CQ"P 90 = = = =NnccimC,P ',Q",P,Q ', P"cngthucmtngtrn.

pdngnhlPascalchosuimC,P ',Q",P,Q ', P" tac:

CP ' PQ ' A,

P 'Q" Q 'P" X,

Q"P P"C B.

= = =

VyA,X,B thnghng.

VyXdichuyntrnngthngABcnh.

Biton5: (Poland1997)Ng gic ABCDE li tha mn:

0CD DE,BCD DEA 90 = = = . im F trong on AB sao

choAF AEBF BC

=

Chngminhrng: FCE ADE,FEC BDC = = .

Ligii:Gi P AE BC = , Q, R ln lt l giao im ca AD v BD vi ng trn ng knh PD,G QC RE = .

pdngnhlPascalchosuim P,C,Q,D,R,E, tac:

PC DR B,

CQ RE G,

QD EP A.

= = =

VyA,G,B thnghng.

Lic:

X

Q"

Q'

P"

P'

A

B C

P

R

Q

P

F

A E

D

C

B

DAG

DBG

DAE

DBC

sinGQDDA GQSAG DG.DA.sinGDQ DA.GQ DA.sinQREDG

BG S DB.GRDG.DB.sinGDR sinGRD DB.sinRQCDB GR

DG

SDA.sin ADE DA.DE.sin ADE AES BCDB.sin BDC DB.DC.sin BDC

AG AFF G

BG BF

= = = = =

= = = =

=

Tddngc FCE ADE,FEC BDC = = .

Biton6:ChotamgicABCnitipngtrn(O),A,B,CltrungimBC,CA,AB.ChngminhrngtmngtrnngoitipcctamgicAOA,BOB,COCthnghng.Ligii:GiA,B,CltrungimcaOA,OB,OC.I,J,KltmccngtrnngoitipcctamgicAOA,BOB,COC.KhiI lgiaoimcacctrungtrccaOAvOA,haychnh lgiaoimcaBCv tip tuyncangtrn(OOA)tiA.TngtviJ,K.p dng nh l Pascal cho su imA",A",B",B",C",C" tac:

A"A" B"C" I,

A"B" C"C" K,

B"B" C"A" J.

=

=

=Vy I, J,K thnghng.

Biton7: (China2005)MtngtrnctcccnhcatamgicABCtheoth

tticcim 1 2 1 2 1 2D ,D ,E ,E ,F ,F . 1 1 2 2 1 1 2 2 1 1 2 2D E D F L,E F E D M,FD F E N = = = .

ChngminhrngAL,BM,CNngquy.Ligii:

K

J

I B"

A"

C"

C' B'

A'

O

B C

A

Z

NM

R

Q

P

L

F2

F1

E2

E1

D2D1

A

B

C

Gi 1 1 2 2 1 1 2 2 1 1 2 2D F D E P,E D E F Q,FE F D R = = = .

pdngnhlPascalchosuim 2 1 1 1 2 2E ,E ,D ,F ,F ,D tac:

2 1 1 2

1 1 2 2

1 1 2 2

E E FF A,

E D F D L,

D F D E P.

=

=

=

Suyra A,L,P thnghng.

TngtB,M,Qthnghng,C,N,Rthnghng.

2 1 1 2 1 2 2 1 1 2 1 2 2 1 1 2 1 2E E D F CA D F X,F F E D AB E D Y,D D FE BC FE Z = = = = = = p dng

nhlPascalchosuim 1 1 1 2 2 2F ,E ,D ,D ,F ,E tac:

1 1 2 2

1 1 2 2

1 2 2 1

FE D F R,

E D F E Q,

D D E F Z.

=

=

=

SuyraQ,R, Z thnghng.

TngtP,Q,Ythnghng,Z,P,Xthnghng.XtcctamgicABC,PQRc:X CA RP,Y AB PQ,Z BC QR = = = .

pdngnhlDesarguessuyraccngthngAP AL,BQ BM,CR CN ngquy.

Biton8: (nhlBrianchon)LcgicABCDEFngoitipmtngtrn.KhiAD,BE,CFngquy.

Ligii:Taschngminhnhlnybngccvicc thy rng Pascal vBrianchon l hai kt qulinhpcanhau.GicctipimtrncccnhlnltlG,H,I,J,K,L.KhiGH,HI,IJ,JK,KL,LGlnltlicccaB,C,D,E,F,A.GiGH JK N,HI KL P, IJ LG=M = =

TheoPascalcholcgicGHIJKLtacM,N,Pthnghng.

NPM

A

B

C D

E

F

H

G

L

K

J

I

MM,N,Pln lt licccaAD,BE,CFnnsuyraAD,BE,CFngquyticccangthngMNP.

Biton9:ChotamgicABC,ccphngicvngcaotinhB,ClBD,CE,BB,CC.ngtrnnitip(I)tipxcviAB,ACtiN,M.ChngminhrngMN,DE,BCngquy.Ligii:GihnhchiucaCtrnBD lP,hnhchiucaBtrnCElQ.Dchngminh:

0ANMI ICP NMI PMI 1802

= = + =

NnM,N,Pthnghng.TngtsuyraM,N,P,Qthnghng.p dng nh l Pascal cho su imB',C ',B,P,Q,C, tac:

B'C ' PQ S,

C 'B QC E,

BP CB' D.

=

=

=VyS,E,D thnghng,haylMN,DE,BCngquytiS.

Biton10:ChotamgicABCnitipngtrn(O).Tiptuynca(O)tiA,BctnhautiS.MtcttuynquayquanhSctCA,CBtiM,N,ct(O)tiP,Q.ChngminhrngM,N,P,Qlhngimiuha.Ligii:VtiptuynME,MDca(O)ctSA,SBtiK,L.pdngnhlNewtonchotgicngoitipSKMLtacBE,AD,SM,KLngquy.pdngnh lPascalchosuim A,D,E,E,B,C,

tac:AD EB I,

DE BC N ',

EE CA M.

=

=

=Vy I,N ',M thnghng,hayN N ' ,tclN DE .

P

Q

S

C'

B'

N

M

IE

D

A

B C

I

L

K

D

E

MN QS

A

B

CP

DoDElicccaMivi(O)nn M,N,P,Qlhngimiuha.

Biton11: (nhlSteiner)ngthngPascalcacclcgicABCDEF,ADEBCF,ADCFEBngquy.Ligii:

Gi 1 1 2AB DE P ,BC EF Q ,AD BC P , = = =

2 3 3DE CF Q ,AD FE P ,CF AB Q . = = =

pdngnhlPascalchosuimA,B,C,F,E,D, tac:

1 3 1 3

1 2 1 2

2 3 2 3

PQ Q P AB FE P,

PQ Q P BC ED Q,

Q Q P P CF DA R.

= =

= =

= =

Vy P,Q,Rthnghng.

pdngnhlDesarguessuyraccngthng

1 1 2 2 3 3PQ ,P Q ,P Q ngquy.

HayngthngPascalcacclcgicABCDEF,ADEBCF,ADCFEBngquy.

Biton12: (nhlKirkman)ngthngPascalcacclcgicABFDCE,AEFBDC,ABDFECngquy.

Tabittrnlc60ngthngPascal.C3ngmtngquytora20imSteiner.Trong20imSteinerc4immt li nm trnmtng thng to ra15ng thngPlucker.Ngoira60ngthngPascal lic3ngmtngquytora60imKirkman.MiimSteinerlithnghngvi3imKirkmantrn20ngthngCayley.Trong20ngthngCayley,c4ngmtlingquytora15imSalmon

ktthcxinaramtsbitonkhcpdngnhlPascal:

Biton13: (MOSP2005)Chotgicni tipABCD,phngicgcActphngicgcBtiE.imP,Q ln ltnm trnAD,BCsaochoPQiquaEvPQsongsongviCD.Chngminhrng AP BQ PQ + = .

R

Q

P

Q3P3

Q2

P2

Q1P1

A F

B

C DE