laisac hinhhocphang.nt
TRANSCRIPT
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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))
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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.
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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
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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
+ + + +
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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