MATHSCOPE.ORG Seeking the Unification of Math Phan c Minh Trng Tn Sang Nguyn Th Nguyn Khoa L Tun Linh Phm Huy Hong Nguyn Hin Trang Tuyn tp cc bi ton HNH HC PHNG Cc bi ton n tp tuyn sinh lp 10 Cc bi ton n tp Olympiad Thng 10/2011 MATHSCOPE.ORG Seeking the Unification of Math Phan c Minh Trng Tn Sang Nguyn Th Nguyn Khoa L Tun Linh Phm Huy Hong Nguyn Hin Trang Tuyn tp cc bi ton HNH HC PHNG Cc bi ton n tp tuyn sinh lp 10 Cc bi ton n tp Olympiad Thng 10/2011 1. Quyn sch c kim duyt v ng bi ban qun tr din n MathScope.org v lti sn ca din n MathScope.org. Cm mi hnh thc sao chp v dn cc logo khnghp l. Cc hnh thc upload file sch ln cc mng x hi, cc trang cng ng, cc dinn khc,. . . u phi ghi r ngun din n MathScope.org.2. Sch c tng hp phi li nhun. Cm mi hnh thc thu li nhun t vic bn, photosch v cc loi hnh khc.3. Sch c tng hp t ngun ti nguyn ca din n MathScope.org. Do sch cquyn khng nu tn cc tc gi ca li gii cc bi ton v ngi bin son chnh sani dung v hnh thc din t sao cho hp l.4. Mi thc mc v bn quyn xin lin h vi ban qun tr din n MathScope.org hoc gitrc tip ln din n.5. Nu bn khng ng vi nhng iu khon nu trn, xin vui lng khng s dng sch.Vic s dng quyn sch chng t bn chp nhn cc iu khon trn.3MclcLiniu 4Ccthnhvinthamgiabinson 5Phnmt.Cckinthccbn 6Phnhai.Tuyntpccbiton 9I.bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.Ccbitonntptuynsinhlp10 . . . . . . . . . . . . . . . . . . . . . . 92.CcbitonntpOlympiad . . . . . . . . . . . . . . . . . . . . . . . . . . 14II.Hngdnvgi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.Ccbitonntptuynsinhlp10 . . . . . . . . . . . . . . . . . . . . . . 212.CcbitonntpOlympiad . . . . . . . . . . . . . . . . . . . . . . . . . . 26III.Ligiichitit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.Ccbitonntptuynsinhlp10 . . . . . . . . . . . . . . . . . . . . . . 382.CcbitonntpOlympiad . . . . . . . . . . . . . . . . . . . . . . . . . . 744LiniuTbuiskhaitrongxhiloingi,tonhclungnlinvicclnhvcisngnhkintrc, hi ha, khoahc,. . . Vtronghuhtcclnhvccatonhc, hnhhcphnglun gi v tr ng u v n chnh l nn tng xy dng nn hnh hc khng gian, l c s caccngnhkintrc,nghthutvtonhcngdng.Cngnhlchsphttrin,chngtatipxcvihnhhcphngtrtsm.Cckhinimvim,ngthng,onthng c cp n ngay tiu hc. Hnh hc tri di n tn nm cui cp THPT v i theonnhngnmi hc, iunykhngnhvai trquantrngcahnhhcni chungvhnhhcphngniring.ngthivisphttrincatonhc,hnhhcphngcngphttrinkhngngng.Lintip cc kt qu mi c pht hin v nhng k thut mi c khm ph. Chnh v th, vicbtkpcckinthccahnhhcphnglcnthitvquantrng. ycngchnhlldoquynschTuyntpccbitonhnhhcphng rai.QuynschctnghpttinguyntrndinnMathScope.orgvltisncaMathScope.org,tcgiccbitonvligii,nhmtnghpulccthnhvincadinnMathScope.orgvimongmuncung cp cho bn hc sinh, sinh vin v thy c gio trn ton quc mt ti liu phong ph vhnhhcphng,htrchoqutrnhhctpvgingdy.Tuyntpccbi tonhnhhcphng khngch nhmvoi tngdthi Olympicmcnlnguntiliuchoccemhcsinhcp2chunbchokthituynsinhlp10.Do,ccbitoncchiathnh2phn:dnhchoccemnthilp10vccbnthiOlympicphhphnvibnc.Mibitonucnhnghngdn,gitrckhinuraligiichititgipbncsuylunvtiptcgiiquytbitonvinhnggi.Xinlurngnhngli nhnxttrongphnhngdnvgi lnhngkinchquancangibinson.XincmnbanquntrvccthnhvindinnMathScope.orgnggp, nghvgiphonthnhquynschny. VxincmnthyChuNgcHng-giovintrngTHPTNinhHi,NinhThunhtrvLATEXhonthinquynsch.Tuynhin, chcchnrngcunschvncnnhnghnchnhtnh, chngti rthoannghnhnhngkinnggp, chiascabnccunschchonthinhn. Bnc c th gp bng cch gi email ring ti hm thalephvn@gmail.com hoc gi trc tiplndinnMathScope.org(http://forum.mathscope.org/index.php).Thaymtnhmbinson,tixinchnthnhcmnsquantmcabnc!HNi,ngy31thng10nm2011idinnhmbinsonChbinPhancMinh5CcthnhvinthamgiabinsonNidungPhancMinh(novae)-HKHTN,HQGHN.TrngTnSang(sang89)-WestminsterHighSchool,California,USA.NguynThNguynKhoa(liverpool29)-THCSNguynTriPhng,ThnhphHu.LTunLinh(conami)-THPTchuynLamSn,ThanhHa.PhmHuyHong(hoangkhtn)-THPTchuyn,HKHTN,HQGHNi.NguynHinTrang(tranghieu95)-THPTchuynPhanBiChu,NghAnHtrkthutLATEXChuNgcHng(hungchng)-GiovintrngTHPTNinhHi,NinhThun.TrnhbybaVAnhKhoa(anhkhoavo1210)-HKHTN,HQGTPHCM.PhancMinh.6Phnmt.Cckinthccbn1.nhlMenelausChotamgicABC, ccimD, E, FtheothtnmtrnccngthngBC, CA, AB.KhiD, E, FthnghngkhivchkhiFAFB

DBDC

ECEA= 1Ch:nhlMenelauscthmrngchoagiclincnh.2.nhlCevaChotamgicABC, ccimD, E, FtheothtnmtrnccngthngBC, CA, AB.KhiAD, BE, CFngquykhivchkhiFAFB

DBDC

ECEA= 13.ngthngEulerChotamgicABC; O, G, Htheothtltmngtrnngoitip,trngtmvtrctmtamgic.KhiO, G, HthnghngvOH= OG.ngthngiquaO, G, HcgilngthngEulercatamgicABC.4.ngtrnEulerVimitamgicABCbtk,9im:trungimcccnh,chnccngcao,trungimcc on thng ni trc tm tam gic vi cc nh cng nm trn mt ng trn, gi l ngtrnEulercatamgicABC. ngtrnEulercbnknhbngmtnabnknhngtrnngoitiptamgicvctmltrungimonthngnitrctmvtmngtrnngoitiptamgic.5.nhlconbmChongtrn(O)vIltrungimcamtdycungAB.QuaIdnghaidycungtyMN, PQsaochoMP, NQctABtiE, Ftheotht.KhiIltrungimEF.6.nhlPtolemyVimitgicliABCDnitiptrongmtngtrn,taucngthcABCD +ADBC= ACBDTngqut:(btngthcPtolemy)VimitgicABCDbtk,tacbtngthcABCD +ADBCACBDngthcxyrakhivchkhiABCDltgiclinitip.77.nhlStewartVibaimA, B, CthnghngvmtimMbtk,tacMA2 BC +MB2 CA +MC2 AB +ABBCCA = 0HaihququenthuccanhlStewartlcngthcdingtrungtuynvdingphngictrong:ChotamgicABC.tBC= a, CA = b, AB= c; ma, lalnltldingtrungtuynvdingphngictrongngvinhAcatamgic.Khitacm2a=b2+c22a24l2a= bc_1 a2(b +c)2_8.ngthngSimsonChotamgicABCvmtimMnmtrnngtrnngoitiptamgic.GiX, Y, ZlnltlhnhchiuvunggccaMtrnccngthngBC, CA, AB.KhiX, Y, ZthnghngvngthngiquachngcgilngthngSimsoncaimMivitamgicABC.Tng qut : Cho tam gic ABCv mt im Mbt k trong mt phng tam gic. Gi X, Y, ZlnltlhnhchiuvunggccaMtrnccngthngBC, CA, AB.KhiiukincnvMnmtrnngtrnngoitiptamgicABClX, Y, Zthnghng.9.ngthngSteinerChotamgicABCvmtimMnmtrnngtrnngoitiptamgic.GiX, Y, Zlnlt l cc im i xng vi Mqua BC, CA, AB. Khi X, Y, Zthng hng v ng thngi qua chng c gi l ng thng Steiner ca im Mi vi tam gic ABC. ng thngSteinerluniquatrctmtamgic.10.imMiquelcatamgic,tgictonphnChotamgicABCvbaimM, N, PtngngnmtrnccngthngBC, CA, AB.KhiccngtrnngoitipcctamgicANP, BPM, CMNngquytiimMiquelXcaM, N, PivitamgicABC.KhiM, N, Pthnghng,tacXimMiquelcatgictonphnABCMNP.KhiXnmtrnngtrnngoitiptamgicABC.11.ngtrnMiquelcatgictonphnCho t gic ton phn ABCDEF, im Miquel Mca t gic v tm ngoi tip cc tam gicAEF, CDE, BDF, ABCcngnmtrnngtrnMiquelcatgic.812.nhlPascalCho6imA, B, C, D, E, Fcngnmtrnmtconicbtk.GiG, H, Ktheothtlgiaoimcacccpngthng(AB, DE), (BC, EF), (CD, FA).KhiG, H, Kthnghng.13.nhlPappusCho hai ng thng a, b. Trn a ly cc im A, B, C; trn b ly cc im D, E, F. Gi G, H, Kln lt l giao im ca cc cp ng thng (AE, DB), (AF, CD), (BF, CE). Khi G, H, Kthnghng.nhlPappusltrnghpsuybincanhlPascalkhiconicsuybinthnhcpngthng.14.BtngthcAM-GMVia1, a2, . . . , anlccsthckhngmtha1 +a2 + +ann

na1a2 anngthcxyrakhivchkhia1= a2== an.15.BtngthcCauchy-SchwarzVia1, a2, . . . , anvb1, b2, . . . , bnlccsthcth_a21 +a22 + +a2n_ _b21 +b22 + +b2n_ (a1b1 +a2b2 + +anbn)2ng thc xy ra khi v ch khia1b1=a2b2==anbn. Trong quy c nu mu bng 0 th tbng0vngcli.16.BtngthcNesbittVia, b, clccsthcdngthab +c+bc +a+ca +b 32ngthcxyrakhivchkhia = b = c.9Phnhai.TuyntpccbitonI.bi1.Ccbitonntptuynsinhlp10Bi 1.1. TamgicABCvungti AcBC=2AB. LyD, EnmtrnAC, ABsaocho

ABD=13

ABCv

ACE=13

ACB.FlgiaoimcaBD, CE.H, KlimixngcaFquaAC, BC.(a) ChngminhH, D, Kthnghng.(b) ChngminhtamgicDEFcn.Bi 1.2. ng trn (O) ni tip tam gic ABC(AB> AC) tip xc vi AB, ACti P, Q. GiR, SlnltltrungimBC, AC.GiaoimcaPQ, RSlK.ChngminhrngB, O, Kthnghng.Bi 1.3.ChotamgicABCnhnnhnHlmtrctm. Chngminhrng, tacbtngthc:HA +HB +HC R.DngcttuynAMNca(O)khngquatm(AM< AN).Chngminhrng(a) ngtrnngoitip OMNluniquamtimcnhH(HkhngtrngO)khicttuynding.(b) TiptuyntiMvNca(O)ctnhautiT.ChngminhTdingtrnmtngthngcnhkhicttuynAMNding.Bi 1.33.Cho ABCc

BAC=60, AC=b, AB=c(b>c). ngknhEFcangtrnngoitiptamgicABCvunggcviBCtiM.IvJlchnngvunggchtExungAB; AC;HvKlchnngvunggchtFxungAB; AC.(a) ChngminhIJ HK.(b) TnhbnknhngtrnngoitiptamgicABCtheobvc.(c) TnhAH +AKtheobvc.Bi1.34. Cho tam gic ABC. Mt im Ddi ng trn cnh BC. Gi P, Q tng ng l tmngtrnnitipcacctamgicABD, ACD.ChngminhrngkhiDdingthngtrnngknhPQluniquamtimcnh.Bi 1.35.ChotamgicABCcphngicADvtrungtuynAM. ngtrnngoi tiptam gic ADMct ABti Ev ACti F. Gi L l trung im EF. Xc nh v tr tng icahaingthngMLvAD.Bi1.36.ChoBCldycungca(O; R).tBC= aR.imAtrncungBCln,kccngknhCI, BK.tS=AB +ACAI +AK .ChngminhrngS=2 +4 a2a.TtmgitrnhnhtcaS.Bi1.37. Cho tam gic ABCni tip (O, R) c

BAC90. Cc ng trn (A; R1), (B; R2),(C; R3)imttipxcngoivinhau.ChngminhrngSABC=BCR21 +ACR22 +ABR23 + 2R1 R2 R34RBi1.38. Cho hnh thoi ABCDc cnh l 1. Trn cnh BCly M, CDly Nsao cho chu viCMNbng2v2

NAM=

DAB.Tnhccgccahnhthoi.Bi 1.39.VphangoicatamgicABCdngcchnhvungBCMN, ACPQctmOvO

.14(a) Chng minh rng khi c nh hai im A, Bv cho Cthay i th ng thng NQ luniquamtimcnh.(b) GiIltrungimcaAB.Chngminh IOO

ltamgicvungcn.Bi 1.40.Chohai ngtrn(O; R)v(O

; R

)ngoi nhaubitOO

=d>R + R

. Mttiptuynchungtrongcahai ngtrntipxcvi (O)ti Evtipxcvi (O

)ti F.ng thng OO

ct (O)tiA, Bv ct (O

)tiC, D(B, Cnm gia A, D).AEct CFtiM,BEctDFtiN.GigiaoimcaMNviADlI.TnhdiOI.Bi 1.41. Cho tam gic ABCc din tch S0. Trn cc cnh BC, CA, AB ly cc im M, N, PsaochoMBMC= k1, NCNA= k2, PAPB= k3(k1, k2, k3< 1).HytnhdintchtamgictobicconthngAM, BN, CP.2.CcbitonntpOlympiadBi 2.1. (APMO2000)ChotamgicABCvi trungtuynAMvphngicAN. ngthng vung gc vi ANti Nct AB, AMln lt ti P, Q. ng thng vung gc vi ABtiPctngthngANtiO.ChngminhrngOQvunggcviBC.Bi 2.2.(DtuynIMO1994)TamgicABCkhngcnti AcD, E, FlcctipimcangtrnnitiplnBC, CA, AB.XlimbntrongtamgicABCsaochongtrnnitiptamgicXBCtipxcviBCtiD,vtipxcviXB, XCtiY, Z.ChngminhrngE, F, Y, Zngvin.Bi 2.3.DnghnhvungDEFGnitiptamgicABCsaochoD, E BC; F AC; G AB. Gi dAl trc ng phng ca hai ng trn (ABD), (ACE). Ta nh ngha cc ngthngdB, dCtngt.ChngminhrngccngthngdA, dB, dCngquy.Bi2.4.ChotamgicABCvitrngtmG.MtngthngdiquaGctBC, CA, ABlnlttiM, N, P.Chngminhrng,tacngthc:1GM+1GN+1GP= 0Bi2.5.ChotgicABCDnitipngtrn(O)ccccnhikhngsongsongvccngchoctnhautiE.FlgiaoimcaADviBC.M, NlnltltrungimcaAB, CD.ChngminhrngEFltiptuyncangtrnngoitiptamgicEMN.Bi 2.6.ChotamgicABCvingtrnnitip(I)vE, Flcctipimca(I)viCA, AB.LyKbtkthuconEF,giH, LlgiaoimcaBK, CKviAC, ABtngng.ChngminhrngHLtipxcvi(I).Bi 2.7. Gi BH, BDlnltlngcaovphngiccatamgicABC. N, L, MlnltltrungimcaBH, BD, AC.LyKlgiaoimcaMNvBD.Chngminhrng,AL, AKlhaingnggictronggc

BAC.Bi 2.8.ChotamgicABCvungti A. TrncctiaAB, AClyE, FtngngsaochoBE= BC= CF. Chng minh rng vi mi im Mnm trn ng trn ng knh BC, taucMA +MB +MCEF15Bi2.9. Cho tam gic ABCc BC= a, CA = b, AB= c v Il tm ng trn ni tip tamgicABC.ChngminhrngIA +IB +IC ab +bc +caBi2.10. T im A nm ngoi ng trn (O), k hai tip tuyn AB, ACn (O). Gi E, FltrungimcaAB, AC.LyDlmtimbtktrnEF,vcctipDP, DQtingtrn.PQctBC, EFlnlttiN, M.Chngminhrng,ON | AM.Bi2.11. Cho tam gic ABCcn ti A ni tip ng trn (O). Trn cnh y BC, ly imM(MkhcB, C).VngtrntmDquaMtipxcviABtiBvngtrntmEquaMtipxcviACtiC.GiNlgiaoimthhaicahaingtrnny.(a) Chng minh rng tng bn knh ca hai ng trn (D), (E) l khng i khi Mdi ngtrnBC.(b) TmtphptrungimIcaDE.Bi 2.12.ChoMlimdingtrnngtrn(O, r)chaingknhcnhAB, CDvunggcvinhau.GiIlhnhchiucaMlnCDvPlgiaoimcaOM, AI.TmtphpccimP.Bi 2.13. ChotamgicuABCvmt imMbt k trongmt phngtamgic. Gix, y, zlkhongcchtMnccnhA, B, Cvp, q, rlkhongcchtMncccnhAB, BC, CA.Chngminhrng:p2+q2+r2

14(x2+y2+z2)Bi 2.14. Cho a gic u A1A2A3A4A5A6A7 v im Mbt k trong mt phng. Chng minhrngMA1 +MA3 +MA5 +M7MA2 +MA4 +MA6Bi 2.15. Tam gic ABCkhng cn ni tip (O) c A1, B1, C1l trung im ca BC, CA, AB.Gi A2lmtimtrntiaOA1saocho2tamgicOAA1vOA2Angdng. CcimB2, C2nhnghatngt.ChngminhrngAA2, BB2, CC2ngquy.Bi 2.16.ChotamgicABCvi MltrungimBC. Vngtrn(O)tyquaAvctcconAB, AC, AMlnlttiB1, C1, M1.Chngminhrng,AB1 AB +AC1 AC= 2AM1 AMBi2.17.ChotamgicABCnitipngtrnbnknhR.GiqlchuvitamgiccccnhltmccngtrnbngtiptamgicABC.Chngminhrng:q63RBi 2.18.ChotamgicABCc: BC=a; CA=b; AB=c; vrvRtheothtlbnknhngtrnnitipvngoitiptamgicABC.ChngminhrngrR+(a b)2+ (b c)2+ (c a)216R2

1216Bi 2.19.ChotamgicABC.CcngphngicBE, CFctnhautiI.AIctEFtiM. ng thng qua Msong song vi BCtheo th t ct AB, ACti N, P. Chng minh rngMB +MC< 3NPBi2.20. Cho tam gic ABCnhn vi ng cao CFv CB> CA. Gi O, Hln lt l tmngoitipvtrctmcatamgicABC.ngthngquaFvunggcviOFctACtiP.Chngminhrng

FHP=

BAC.Bi2.21.Chongtrn(O; R)vmtimPcnhbntrongngtrn.AB, CDl2dycungdingca(O)nhngluniquaPvlunvunggcvinhau.(a) ChngminhrngPA2+PB2+PC2+PD2khngi.(b) GiIltrungimBC.HiIdingtrnngno?Bi2.22.ChotamgicABCvimMbtknmtrongtamgic.Chngminhrng:MA +MB +MC + minMA, MB, MC < AB +BC +CABi 2.23.TamgiccnABCni tip(O)cAB=ACvAQlngknhca(O). LyM, N, Pln lt trn cnh AB, BC, CA sao cho AMNPl hnh bnh hnh. Chng minh rngNQ MP.Bi2.24.ChotgicABCDcM, NlnltltrungimAB, CDvOlgiaoimca2ngcho.GiH, KltrctmcatamgicOAB, OCD.HychngminhMN HK.Bi 2.25. Cho t gic ABCD ni tip (O) c hai ng cho ct nhau ti I. Gi M, Nln ltltrungimcaAB, CD. P, QlchnngcaoktIcatamgicIAD, IBC. Chngminhrng,PQ MN.Bi2.26. Cho tam gic ABCv tam gic DBCc tm ni tip ln lt l H, K. Chng minhrngADHK.Bi 2.27.ChoKlimnmtrongtamgicABC. MtngthngquaKcthai cnhAB, ACtheothtM, N.Chngminhrng:SABC8_SBMKSCNKBi 2.28. Cho tam gic ABC nhn v Ml mt im thuc min trong tam gic. Gi A1, B1, C1lnltlgiaoimcaMA, MB, MCvi cccnhtamgicABC. LyA2, B2, C2lccim i xng vi Mqua trung im ca B1C1, C1A1, A1B1. Chng minh rng AA2, BB2, CC2ngquy.Bi2.29.ChotamgicABCnitip(O; R)cMthuccungBCkhngchaA.TmvtrcaMP= 2010MB + 2011MCtgitrlnnht.Bi 2.30. ChotamgicABC. CcimD, E, FnmtrncccnhBC, CA, ABsaochoAD, BE, CFngquytiO.QuaOkngthngsongsongviBCctDE, DFtheothttiHvK.ChngminhOltrungimHK.Bi 2.31. ChotamgicABC. Mlmt imbt k trnmt phngvkhngnmtrn17tam gic ABC. Cc ng thng AM, BM, CMln lt ct cc ng thng BC, CA, ABtiD, E, F.GiH, KlnltlgiaoimcacccpngthngBMviFD;CMviED.ChngminhccngthngAD, BK, CHngquy.Bi2.32.ChotgicliABCD.Chngminh:minAB, BC, CD, DA AC2+BD22 maxAB, BC, CD, DABi 2.33.Chongtrn(O; R)vhaiimA, BcnhixngvinhauquaO.GiMlimchytrn(O).ngthngMA, MBct(O)tiP, Qtngng.ChngminhrnggitrbiuthcMAAP+MBBQkhngikhiMdichuyntrn(O).Bi2.34. Cho (O)v dy AB.im Mdichuyn trn cung ln AB.Ccng cao AE, BFca ABMctnhautiH.K(H; HM)ctMA, MBCvD.ChngminhngthngktHvung gcviCDlun iqua mt imc nh khiMdichuyn trn cung lnAB.Bi2.35. Cho tam gic ABCni tip ng trn (O). G l trng tm tam gic. AG, BG, CGlnltct(O)tiA1, B1, C1.Chngminhrng:GA1 +GB1 +GC1GA +GB +GCBi2.36. Cho ABCv D, E, Fln lt l hnh chiu ca A, B, Cxung ba cnh tng ng.ng thng qua Dsong song vi EFct AB, ACti P, Q. Bit EF BC= R. Chng minhrngngtrnngoitip PQRiquatrungimBC.Bi 2.37.Chotgicli ABCDni tipngtrn(O). ChoAB=a, CD=b,

AIB=,trongIlgiaoimcahai ngchoACvBD. Tnhbnknhngtrn(O)theoa, bv.Bi 2.38.Cho ABCctrctmH.ngtrnquaB, CctAB, ACtiD, E.GiFltrctm ADEvIlgiaoimcaBEvCD.ChngminhrngI, H, Fthnghng.Bi 2.39. Cho ABC khng cn, ngoi tip ng trn (I). Tip im ca (I) trn BC, CA, ABlnltlD, E, F. DEctABP. MtngthngquaCctAB, FElnltN, M.PMctACQ.ChngminhrngINvunggcviFQ.Bi 2.40.ChotgicABCD.GiI, JtheothtltrungimcaAC, BD.Chngminhrng:AC +BD + 2IJ< AB +BC +CD +DABi 2.41. Cho ABCni tipngtrn(O). EthuccungBCkhngchaAvkhngtrngB, C. AEcttiptuynti B, Cca(O)ti M, N. Gi giaoimcaCMvBNlF.ChngminhrngEFluniquamtimcnhkhiEdichuyntrncungBCkhngchaA.Bi 2.42.ChotgicABCDnitipthamnABCD=ADBC.ngtrn(C)quaA, Bvtipxcvi BC, ngtrn(C

)quaA, DvtipxcCD. ChngminhrnggiaoimkhcAca(C)v(C

)ltrungimBD.Bi2.43. Cho tam gic nhn ABC, gi Hl trc tm ca tam gic. Tm iu kin cn v 18i vi cc gc ca tam gic 9 im : chn cc ng cao ca tam gic, trung im cc cnhcatamgic,trungimcconthngHA, HB, HClnhcamtagicu.Bi 2.44. Cho tamgic ABC. ng trn(I) ni tiptamgic ABCv tipxc viBC, AC, ABlnltti D, E, F. ChngminhrngID, EFvtrungtuynAM(M BC)ngquy.Bi 2.45.Chohai onthngABvA

B

bngnhau. PhpquaytmMbinAthnhA

,binBthnhB

.PhpquaytmNbinAthnhB

,binBthnhA

.GiSltrungimcaAB.ChngminhrngSMvunggcviSN.Bi 2.46. Cho tam gic ABC, Ml im nm trong tam gic. AM, BM, CMct BC, CA, ABtheo th t D, E, F. Gi H, I, Ktheo th t l hnh chiu ca Mtrn BC, CA, AB. K hiuP(HIK)lchuvitamgicHIK.Hychngminh:P(DEF)P(HIK)Bi 2.47.TamgicABCnhnni tip(O), ngcaoAHct(O)ti A

. OA

ctBCtiA

.XcnhtngtchoB

, C

.ChngminhAA

, BB

, CC

ngquy.Bi2.48.Chongtrn(O)vmtngthngdcnh.GiHlhnhchiucacaOtrnd.LyMcnhthucngtrn.A, BthayitrndsaochoHltrungimAB.GisAM, BMct(O)tiP, Q.ChngminhPQluniquamtimcnh.Bi2.49. Cho ng trn tm Ini tip tam gic ABCtip xc vi BC, AB, ACti D, E, F.QuaEvngsongsongviBCctAD, DFM, N.ChngminhrngMltrungimcaEN.Bi 2.50.ChotamgicABCcAB= c, BC= a, AC= bvIltmngtrrnnitip.Hai im B

, C

ln lt nm trn hai cnh AB, ACsao cho B

, C

, Ithng hng. Chng minhrngSABC a +b +c2bc

_SAB

CSABC

Bi2.51. Cho t gic ABCDni tip. E, F, G, Hln lt l tm ng trn ni tip cc tamgicABC,BCD, CDA, DAB.ChngminhrngtgicEFGHnitip.Bi 2.52.ChohnhvungABCD. ItythucAB, DIctBCti E, CIctAEti F.ChngminhrngBF DE.Bi 2.53.ChotamgicABCkhngvungnitipngtrn(O),trctmH.dlngthng bt k qua H. Gi da,db, dcln lt l cc ng thng i xng vi d qua BC, CA, AB.Chngminhrngda,db,dcngquytimtimtrn(O).Bi2.54. Cho hnh thang ABCD (AB | CD). ACct CDti O. Bit khong cch t OnADvBCbngnhau,hychngminhrngABCDlhnhthangcn.Bi2.55.ChotamgicABCcntiA.ngtrntipxcAB, AC,ctBCtiK.AKcttiimthhailM.P, QlimixngcaKquaB, C.ChngminhrngngtrnngoitiptamgicMPQtipxcvi.Bi2.56. Cho tam gic ABCvung ti A c B= 20, phn gic trong BI. im Hnm trn19cnhABsaocho

ACH= 30.Hytnhso

CHI.Bi2.57.ChotamgicABCngoitip(I).GiD, E, FlnltlimixngviIquaBC, CA, AB.ChngminhrngAD, BE, CFngquy.Bi2.58. Cho tam gic ABCcn ti A ni tip (O). im Ml trung im ca AC. BMctli(O)tiimthhailQ.Chngminhrng2AQBQ.Bi2.59. Cho ABCtha mn AB +BC= 3CA. ng trn ni tip (I) tip xc AB, BCtiD, E.GiK, LtngngixngviD, EquaI.ChngminhrngtgicACKLnitip.Bi 2.60.ChotamgicABCngoi tip(I). (I)tipxcBC, CA, ABlnltti D, E, F.ChngminhrngtmngtrnngoitipcctamgicAID, BIE, CIHthnghng.Bi2.61.ChotamgicABCnitip(O).M, NlnltlimchnhgiacungABkhngcha Cv cung ACkhng cha B. Dl trung im MN. G l mt im bt k trn cung BCkhngchaA. Gi I, J, Klnltltmni tipcctamgicABC, ABG, ACG. LyPlgiaoimthhaica(GJK)vi(ABC).ChngminhrngP DI.Bi2.62.ChongicuA1A2. . . An(n 4)thamniukin1A1A2=1A1A3+1A1A4Hytmn.Bi 2.63. Gi AA1, BB1, CC1tngnglccngphngictrongcatamgicABC.AA1, BB1, CC1ctngtrnngoitiptamgictiA2, B2, C2theotht.Chngminhrng:AA1AA2+BB1BB2+CC1CC2

94Bi2.64.ChotamgicABC,ngthngdctccngthngBC,CA,ABlnlttiD, E, F.GiO1, O2, O3lnltltmngtrnngoitipcctamgicAEF, BDF, CDE.ChngminhrngtrctmtamgicO1O2O3nmtrnd.Bi 2.65.ChotgicABCD,ACctBDtiO.GiM, N, P, QlnltlhnhchiucaOtrnAB, BC, CD, DA.BitrngOM= OP, ON= OQ.ChngminhrngABCDlhnhbnhhnh.Bi 2.66.ChotamgicABC, phngictrongAD(D BC). Gi M, Nlccimthuctia AB, ACsao cho

MDA =

ABC,

NDA =

ACB. Cc ng thng AD, MNct nhau ti P.Chngminhrng:AD3= ABACAPBi 2.67.Trnmtphngcho2000ngthngphnbit, i mtctnhau. Chngminhrngtntitnht2ngthngmgccachngkhnglnhn1802000().Bi 2.68.ChotgicABCDni tip(O)cAB=AD. M, NnmtrncccnhBC, CDsaochoMN= BM+DN.AM, ANct(O)tiP, Q.ChngminhrngtrctmtamgicAPQnmtrnMN.Bi 2.69.ChotgicABCD. Hai ngchoAC, BDctnhauti O. Gi r1, r2, r3, r4ln20ltlbnknhccngtrnnitipcctamgicAEB, BEC, CED, DEA.Chngminhrng1r1+1r3=1r2+1r4liukincnvtgicABCDngoitipcmtngtrn.Bi2.70. Cho tam gic ABCc Ml trung im ca BCv Hl trc tm tam gic. ngthngvunggcviHMtiHctAB, ACtiD, E.ChngminhrngHltrungimcaDE.Bi2.71.ChoonthngAB= acnh.imMdingtrnAB(MkhcA, B).TrongcngmtnamtphngblngthngABdnghinhvungAMCDvMBEF. HaingthngAF, BCctnhauN.TmvtrimMsaochoonMNcdilnnht.Bi 2.72.ChotamgicABCnhnkhngcn, ni tip(O). CcngcaoAA0, BB0, CC0ngquyti H. Cc imA1, A2thuc (O) saochongtrnngoi tipcc tamgicA1B0C0, A2B0C0tipxctrongvi(O)tiA1, A2.B1, B2, C1, C2xcnhtngt.ChngminhrngB1B2, C1C2, A1A2ngquytimtimtrnOH.Bi 2.73. Chongtrn(I)ni tiptamgicABCtipxcBC, CA, ABti A1, B1, C1.Cc ng thng IA1, IB1, IC1tng ng ct cc on thng B1C1, C1A1, A1B1ti A2, B2, C2.ChngminhccngthngAA2, BB2, CC2ngquy.Bi 2.74.ChotamgicABCcntiA.TrntiaicatiaCAlyimE.GiaoimcaBEv phngicgc

BAClD.Mtng thngquaDsong songABctBCF.AFctBEtiM.ChngminhrngMltrungimBE.Bi 2.75.ChotgicliABCDsaochoABkosongsongviCDvimXbntrongtgictha

ADX=

BCX< 90v

DAX=

CBX< 90.GiY lgiaoimngtrungtrccaABvCD.Chngminhrng

AY B= 2

ADX.Bi 2.76. Cho t gic li ABCD ni tip trong (O). AD ct BCti E, ACct BD ti F.M, NltrungimAB, CD.Chngminhrng:2MNEF=ABCD CDABBi2.77.ChotgicABCDnitipcmtngtrn.Chngminhrng:ACBD=DAAB +BCCDABBC +CDDABi 2.78. Cho tam gic nhn ABCni tip (O; R).Gi R1, R2, R3 tng ng l bn knh ngtrnngoitipcctamgicOBC, OCA, OAB.Chngminhrng:R1 +R2 +R33R21II.Hngdnvgi1.Ccbitonntptuynsinhlp10Bi1.1.(a)Tac

FHD = 20,viccnlichlkimtra

FHK= 20.(b)GiIlgiaoimcaHK, BC.Lnltchngminhccktqusau

DFI= 120BEFInitip

EFI= 120v

FIE= 20=

DIF DFI= EFIKtqucuichngttamgicEFDcntiF.Bi1.2.VichrngSK= SQ,sdngccbinidionthngchrarngRK= RB.Bi1.3.QuaHdngccngthngsongsongvi cccnhtamgicvccgiaoimi vi cccnh cn li. Hy ch cc hnh bnh hnh to c v s dng bt ng thc tam gic, ta sciucnchngminh.Bi1.4.(a)Thai tamgicngdngANB, CPDsuyra

ANBkhngi. TrtracqutchimN.(b)imcnhcntmchnhlgiaoimtiptuyntiA, BcaO.Bi1.5.HychngminhrngB

ltmbngtiptronggcBcatamgicAA

BvC

ltmbngtiptronggcCcatamgicAA

Ctsuyra

B

A

C

= 90.Bi1.6.GiSlgiaoimcaEM, CD.pdngnhlMenelauschohaitamgicACN, BCNvnhlThalesrtra:BC2NC2=KBKNngthcnychngttamgicvungBCNnhnKlmchnngcaoktC.Bi1.7.(a)Bngtnhchtcatiptuynvccphpbini gc, hychngminh

BAE=

BEA.TsuyraNltrungimAEvO, N, Pthnghng.(b)Hychngminh

MDN= 90.(c)ChngminhtgicOKPAnitip.Bi1.8.HychngminhA1B1HaHblhnhbnhhnhnhbsau:VitamgicXY Z,trctmQthQX= Y Zcot X.22Bi1.9.GiNltrungimcaAB.ngtrncnhcntml_N, AB2_.Bi1.10.chngminhktqucabi ton, tasch rarngOSlphngiccagc

CODbngcchsdngcctamgicngdngvtgicnitip.Bi1.11.Hai(a)v(b)ulnhngktqunginvquenthuc.Vi(c),taschngminhAH= 2HI,saupdnghthclngtrongtamgicvungABC.Bi1.12.Bngcchbini gcdavocctgicni tip, hychngminhrngIKlphngictrongcagcDIE.Bi1.13.(a)HychngminhBHCA

lhnhbnhhnh.(b) Thc cht ylkt ququenthuc v ngthngEuler : H, O, GthnghngvHG = 2OG.Bi1.14.DngthmhnhbnhhnhABMT. ThypdngbtngthcPtolemychotgicAMDTvichcconthngbngnhausuyraiucnchngminh.Bi1.15.(a)Hychngminh(O3)ltrctmca AO1O2.(b)Davocctamgicngdng,tasuyrangthcO1HO2H=BHAH=ABACTsuyra O1HO2BAC.(c)Sdngktqusau___R3=AB +AC BC2R2=AH +CH AC2R1=AH +BH AB2Bi1.16.(a)C2cchchngminhcbnnhtchoktquny:VtiptuynCxcaO.HychngminhrngtiptuynnysongsongviEF.VngknhCC

,gigiaoimcaCC

, EFlQ.HychngminhBFQC

nitipsuyraktqu.(b)Suyratrctipt(a).(c) Nhn xt CA2+CB2khng i nh gi chu vi v din tch ABC. Ngoi ra, cn mt23cchnginhnnhgidintchnhvotnhcht: di ngtrungtuyntamgickhngnhhndingcaoxutphtcngmtnh.(d)KhiCdingtrncungABthIlundingtrncungchagc135dngtrnonOAhocOBnmtrnnamtphngbABchaC(trhaiimAvB).Bi1.17.(a)TrntiaCDlyimTsaochoAT= AC.HychngminhCK CF= CT.(b)I BDcnh.(c)pdngngthcEK=AE2DEsuyraonEKngnnhtkhiE C.Bi1.18.(a)Chngminhtuntccngthcsau:EA = EB +EC1ED=EAEBEC(b)pdngngthcchngminh(a).(c)GidicccnhtamgicuABCla.Hychngminhrng:R1 +R2=(3a 2AD)R3aBi1.19.(d)GiIlgiaoimcaAT, BM.Khi,chngminhtunt:MltrungimBI.SNMB=TNTM=ANMIBi1.20.(a)DngMI1 BEtiI1.HychngminhM, I1, Nthnghng.(b)T(a). hychngminhAM+ CN=MNvsuyragitr lnnhtcaSDMNtckhiE D.Bi1.21.Gi I, Klnltltmcaccngtrn(CDE), (ABC). DngngknhCPca(I).Chngminhtuntccktqusau:PM CMPO CMM, O, PthnghngBi1.22.Chngminhtuntccktqusauy: FCDDAE24 ACFEAC ACMAFCAMAF= AD2Bi1.23.ChrngADBCltgiciuha,hytmccngthcvtsdionthngc BDIBCA.Tsuyraiucnchngminh.Bi1.24.(a)HychngminhINDMnitip.(b) Chng minh PN | AB, PM | AC. T suy ra t gic PNQMni tip v c tng 2 gcil180.Bi1.25.GiHltrungimBC,NdingtrnngthngvunggcviAHtiAcnh.Bi1.26.LyNtrnBCsaocho

BAM= 90.pdngcngthcngphngictnhdiANtheoAM, b; AMtheoAN, a.TrtraquanhgiaAMvia, b.Bi1.27.DngtamgicAMEu(EnmtrongtamgicADM).TsuyraDM= DA = DC.ps: MCDu.Bi1.28.(a)Gi HlgiaoimcaKPvIN. HychngminhtgicMNPQchai ngchovunggcvinhautitrungimcamingsuyraiuphichngminh.(b)GiElgiaoimcangtrnngoitiptamgicABKviIK.Chngminhtuntccngthcsau:IDIC= IEIKKBKC= KEIK(c)GiRlgiaoimcaAJ, OL.KAS BO (S BO).Lnltchngminh:JltrungimBS OLFAJBAFROnitipAJ OLBi1.29.Bitonnylhqutrctipcanhlconbm.HychngminhrngMngthiltrungimcacconthngM1M3vM2M4Bi1.30.25pdngcngthcdingtrungtuynchocctamgicACE, ABD, BCD.Bi1.31.(c)GiMltrungimBCthEFluniquaMcnh.(d)SAEIF max SABC max.Bi1.32.(a)ngtrnngoitip OMNluniquaimH AOcnh.(b)TlundingtrnngthngvunggcviOAtiHcnh.Bi1.33.(a)HychngminhccktquAEIJAE | HK(b)R =_b2+c2bc3(c)rng BHF= CKF.ps:IH +IK= b +c.Bi1.34.imcnhcntmchnhltipimcangtrnnitiptamgicABCviBC.ccktquny,tacnsdngbsau:B.Chohai ngtrn(O1), (O2)khngctnhau, hai tiptuynchungtrongd1, d2cttiptuynchungngoidtiA, B.GiC, Dlnltltipimca(O1), (O2)vid.Khi,AC= BD.Bi1.35.Nu ABCcntiAthML AD.NuAB ,= AC,hychngminhBE= CF.TsuyraML | AD.Bi1.36.pdngnhlPtolemychotgicnitipAIBK.Sau,davoa2,hychngminhrng:S=2 +4 a2a 1Bi1.37.t p =a +b +c2, suy ra R1= p a, R2= p b, R3= p c. ng thc cn chng minh tngngvi:a(p a)2+b(p b)2+c(p c)2+ 2(p a)(p b)(p c) = abc chng minh ng thc ny, c th dng phng php khai trin rt gn hoc dng phngphpathc.Phnchngminhdnhchobnc.Bi1.38.DngvphabADkhngchaCtamgicADGsaocho ADG= ABM. HychngminhrngN, D, GthnghngsuyrarngABCDlhnhvung.26Bi1.39.(a)GiLltrungimcaQN.Hychngminh ALBvungcnsuyraLcnh.(b)ChngminhOI, O

Ivunggcvbngnhau.Bi1.40.immuchtcabitonlchngminhMNAD.Tsuyra BINCIM.ps:OI=d2+R2R22d.Bi1.41.ChngminhngthcSBFC=k21 +k2 +k2k3 S0ps:S= S0

(k1k2k31)2(k1k2 +k1 + 1)(k2k3 +k2 + 1)(k3k1 +k3 + 1)2.CcbitonntpOlympiadBi2.1.Davonhngquanhvunggccgithitvquanhvunggccnchngminh,tacthsuynghtheocchngsau:avohtrcta:Ttnhinv2trctaphivunggcvinhau,dotmtanntPhocN. Tuynhin, doNlchnngphngictrongcatamgicABCnnvicttmtiNsthuntinhn.Davotngtrctm: TacOA QN,hytmcchdngtmKsaochoQltrctmcatamgicAOK.TcchdngimK,giibitonngcchngminhrngQchnhltrctmcatamgicAOKtheocchdng.Sdngvector: Sdngvectorlmtphngphpcslachnphongph. Ttnhinngthccnchngminhphil OQ BC=0.Ccvector OQ,BCcthbiudinthnhrtnhiutngcaccvectorkhcnhau.yvalimmnhcngchnhlimyucavector,taphitmnhngcpvectorthchhpcthtnhton.Dnhin BCnncginguyn, OQcthtchthnhtngca2vector OP,PQv 2vector ny u c th tnh c module theo di cc cnh v cc gc ca 2 vector nyhpviBCcngcthxcnhtheoccgccatamgicABC.Bi2.2.HychngminhrngEF, Y Z, BCngquysuyraktqu.Bi2.3.Hy biu din t sMBMCqua cc yu t lin quan n tam gic ABCnh tnh cht ca phngtch.SausdngnhlCevachotamgicABCsuyraiuphichngminh.Bi2.4.ChiuM, N, PtheophngsongsongviBClnngtrungtuynxutphttAcatam27gicABCahthccntnhtonlnngtrungtuyn.Bi2.5.chngminhSEltiptuynca(EMN)mtmngtrnnychaxcnh,tac2hngcbnsauy:Chngminhhthcvgc: Quyvchngminh

FEM=

ENM. Hydngcchnhbnh hnh AEBL, CEDK, tn dng cc tam gic ng dng rt ra ng thc v gctrn.Chngminhhthcvcnh: GisMNctFEtiP(dthyrngPcngchnhltrungimcaEF),tacnchngminhPE2= PM PN.GigiaoimcaAB, CDl S, hy s dng cc nh l v hng im iu ha chng minh ng thc trn. Phncnlixindnhchobnc.Bi2.6.Thcchtylbitonocabquenthuccatgicngoitipngtrn:Ccngchovccngthngni cctipimcangtrnni tipmttgicngoitiplncccnhicatgicngquytimtim.Bi2.7.Hychngminhngthcsau:KDKB

LDLB=AD2AB2ng thc trn chng t AK, AL l hai ng ng gic trong gc BAC. Hy s dng nhlMenelausvchticctrungimtnhton,rtrangthctrn.Bi2.8.Hychn2ngthcsau:aMA = bMB +cMCa2= MB2+MC2S dng 2 ng thc trn v bt ng thc Cauchy - Schwarz, ta suy ra iu cn chng minh.Bi2.9.Hychb:IA =_bc(b +c a)a +b +cT,tacthabitonvbtngthcisnginhn.Bi2.10.tngchnhcabi tonlchngminhAM, ONcngvunggcvi AD. Sauyl2hngcnchtipcnktquny:Ccvicc.Phngtchcamtimvingtrn(O)vvingtrnimtmA.28Bi2.11.(a)GiKlgiaoimcaBD, CE.HysdngnhlThaleschngminhrngR(D) +R(E)= BK= CK.(b)dontrcqutchcaI,tachn3vtrMkhcnhau.TchotagithuytIdingtrnngthngcnhsongsongviBC.Cngchnhtychotatnghng thng vung gc IHxung BC. H vung gc tng t cho D, Exung BC, bng mtsbctnhton,tasthycdionIHkhngi,tsuyraqutchimI.Bi2.12.Cuhnhngtrnvi 2ngknhcnhvunggcvi nhaulmtalintngngaynhtrcta.NuchnA(r, 0), B(r, 0), C(0, r), D(0, r)thqutchcaimPslngcongcphngtrnhy2= 2xr +r2.Bi2.13.GiA

, B

, C

lnltlhnhchiuvunggccaMlnccngthngBC, CA, ABtheotht.Tachngminhccbtngthc,ngthcsausuyraiucnchngminh:p2+q2+r2

13_B

C2+C

A2+A

B2_B

C2+C

A2+A

B2=34_x2+y2+z2_Bi2.14.pdngnhlPtolemychocctgic:MA1A2A3MA5A6A7MA2A4A6A1A3A4A5Kthpvimtsbinihpl,tascngaybtngthccnchngminh.Bi2.15.Trctin,hychngminhrngA2chnhlgiaoimcahaitiptuynktB, Cca(O)vtngtiviB2, C2.Taavbitonquenthucvcthlmtheohaicch:Ta c th thy ngay AA2, BB2, CC2chnh l cc ng i trung ca tam gic ABCnnchngngquytiimLemoinecatamgicABC.p dng nh l Ceva. Tht vy, do (O) tr thnh ng trn ni tip tam gic A2B2C2nnA, B, Ctrthnhtipimcangtrnni tiptrncc cnhtamgicA2B2C2. T, tacthpdngnhlCevachotamgicA2B2C2chngminhAA2, BB2, CC2ngquy.Bi2.16.Ta s a AB1 AB, AC1 AC, AM1 AMthnh cc biu thc cha AB, BC, CA, TB/(O), TC/(O),TM/(O). T bin i ng thc cn chng minh v mt ng thc ng theo cng thc trung29tuyn.Bi2.17.Hychngminhhaibsauy:TamgicXY ZnitipngtrnbnknhRth:XY+Y Z +ZX33RNu Ia, Ib, Icl cc tm bng tip ca tam gic ABCth bn knh ng trn ngoi tiptamgicIaIbIcbng2lnbnknhngtrnngoitiptamgicABC.Bi2.18.immuchtcabitonlbtngthcsauy:R22Rr = OI2 DM2=(b c)24TrongDltipimcangtrnnitiptamgicABCviBCvMltrungimcaBC.Bi2.19.Bitondatrnbsauy:B. Gi H, I, Kl hnh chiu ca im M(c nh ngha trong bi) ln BC, CA, ABthMH= MI +MK.Phncnli lsdngbtngthctamgickhai thcbny. Tasthucbtngthccnchngminh.Bi2.20.Ly Ki xng vi Hqua AB. ng thng PFct (O), BKti M, N, Q. Hy s dng nhlconbmchotamgicABCchngminhPKQHlhnhbnhhnh.Bi2.21.(a)ylmtktqurtquenthuc:PA2+PB2+PC2+PD2= 4R2Mt cch nhanh nht l v ng knh AKca (O) v ch BCDKl hnh thang cn suyraktqu.(b)GiMltrungimcaOP.TrchthychngminhrngIO2+ IP2khngi,tysuyraIdingtrn_M, 2R2OP24_cnh.Bi2.22.Hychngminhvsdngktqusau: Vi imMbtk nmtrongtgicABCD, talunc:MC +MD < DA +AB +BCTrlibiton,hygitrungimcccnhBC, CA, ABkhaithcktqutrn.Bi2.23.chngminhQN MP,tachaihngsau:30Gi Klimi xngcaNquaMP. TaschngminhK (O). TsuyraN, K, Qthnghng.VichrngAK | MP.Tasciucnchngminh.Sdngvector:Phntch QNthnhtngca QB,QC; MPthnhtngca MA,MNv ch ccng vung gcvi nhau. cho tin cho vic bin i,nn tk =NCBC.Bi2.24.Trcht,cnhnxtrng2MN= AC +BD.Tnhnxtny,nugix, ylnltldihnhchiucaHKlnAC, BD;tachcnchngminhxAC= yBD.Bi2.25.Tachaihnggiiquyt:GiKltrungimAC,hychngminhrng KMNIQPsuyraktqu.Sdngvector : Trcht, cnhnxt rng2MN= AC+ BD. Tnhnxt ny,nugi x, ylnltldi hnhchiucaPQlnAC, BD; tach cnchngminhx AC= yBD. V ng thc ny c th chng minh da vo tnh cht phng tch caimIvi(O).Bi2.26.Hychnhaibsau:B 1 : Cho tam gic ABCv mt im Mnm trong tam gic y. Khi MB+MC b).Tabtngthccnchngminhvmtbtngthcisngin.Bi2.59.GiGlgiaoimcaCK, AB; FlgiaoimcaAL, BC; MlgiaoimcaAL, CK.Mtsktqucnchsuyraktluncabiton: AGCcntiA.M (I).Bi2.60.Gi A1, B1, C1lnltltrungimEF, DE, DF. Khi , hyxtphpnghchotmIphng tch k = r2(vi rl bn knh ng trn ni tip tam gic ABC) v ch DA1, BE1,CF1ngquy,tasciuphichngminh.Ngoi ra, ta c th s dng nh l Menelaus. Tuy nhin, ta khng th s dng nh l Meneleuschngminh3tmngoi tipythnghngmtcchtrctip. Thnhng, ch cnrngnugi A2, B2, C2lchnngphngicngoi tamgicABCth tmngoi tipcctamgicAID, BIE, CIFchnhltrungimcaIA2, IB2, IC2. BngnhlMenelaus, dthyrngA2, B2, C2thnghng,tsuyraiuphichngminh.Bi2.61.Gi P

lgiaoimcaDI vi (O)(P

thuccungBCkhngchaA). Khi , hychngminhrng:PMPN=AMAN=P

MP

NngthcnychngtAMPN, AMP

Nultgiciuha.ViunycngchngtP P

.Bi2.62.tx =n_0x 4_.Sdngnhlhmssinccphngtrnh1sin x=1sin 2x+1sin 3xCngviccnlichlgiiphngtrnhtrn_0, 4_.ps:Bitoncnghimduynhtn = 7.Bi2.63.36Hytnhtoncctstrongbitheodicccnhtamgicabtngthccnchngminhvmtbtngthcis.Bi2.64.Gi MlimMiquel catgictonphnBCEFAD. Hychngminhrngdlngthng Steiner ca Mi vi (O1O2O3) suy ra iu cn chng minh (ch ng trn MiquelcatgictonphnvtnhchtcangthngSteiner)Bi2.65.Sdngphnchngchngminh:Bquatrnghptntimtcpcnhisongsong,xttrnghpchaicpcnhiusongsong.Khi,giElgiaoimcaAD, BC; FlgiaoimcaAB, CD. Hychngminhrng, nuABCDkhnglhnhbnhhnhthFO | EO,iunyhinnhinvl.Bi2.66.ngthccnchngminhcsuyrat4ngthcsau:AD2= ANACAD2= AMABAMAD = APACANAD = APABBi2.67.Hytnhtinccngthngchovmtimvchrnggccachngvncboton.pdngnguynlDirichlettasciucnchngminh.Bi2.68.Ly im Htrn on MNsao cho MH= BM, NH= DN. Hy chng minh Hi xng viBquaAP, i xngvi DquaAQ. TsuyraAHPQ, QHAPciucnchngminh.Bi2.69.tAB= a, BC= b, CD= c, DA = d, OA = x, OB= y, OC= z, OD= t.Hytmcchloibccilngx, y, z, ttrongngthccgithit.Tacnbinitngngchcuisla +c = b +d.Khi,pdngnhlPithot,tascABCDngoitip.Bi2.70.Chaicchtipcnbiton:Cch 1 :Ch hai cp tam gic ng dng ADHCHMv AHEBMH. SauhysdngcccptlvcnhcahaicpngdngchngtHE= HD.Cch2: Sdngtnhchtcatskp,hychngminhktqutngqut:MBMC=HDHEBi2.71.Chaihngtipcnbiton:37Cch1: ChngminhNMlphngic

ANBtsuyraMN BC2(phngicnhhntrungtuyn).Cch2: Hychngminh:1MN=1DM+1MENhnxtrngDM+MEkhnginhgiMN.Bi2.72.Gi XAlgiaoimcaBC, B0C0, nhnghatngtchoXB, XC. HychngminhrngXA, XB, XCl cc ca A1A2, B1B2, C1C2. T suy ra rng kt lun ca bi ton tng ngviXA, XB, XCthnghngvngthngiquachngvunggcviOH.Bi2.73.AA2, BB2, CC2chnhlccngtrungtuyncatamgicABC.Ngoi ra, ta cng c th s dng nh l Ceva dng sin chng t AA2, BB2, CC2ng quy.Bi2.74.Bitoncthcgiiquyttheohaicchsau:Cch1: pdngnhlMenelauscho BCEvi ccimA, F, M(saukhi tnhcctsmtcchthchhp).Cch2: GiHltrungimBC.HychngminhrngMH | CE.Bi2.75.Sdngbsau:Chohaingtrn(O1)v(O2)ctnhautiX, Z.LyAlmtimbtknmtrn(O1).DngtiaZBixngtiaZAquaZXviBthuc(O2).GiOltmngoitip ABZ.KhitacOO1= OO2.Bi2.76.Gi PltrungimEF. LyUlimi xngcaFquaN, V ltrungimEU. Hychngminhccktqusau: EBFEDU, PABV CDPMAB=V NCD=PFCD2PNEF=CDABBi2.77.Cchnhanhnhtlsdnghthclinquangiacccnh,dintchvbnknhngoitiptamgic.Tuy nhin, i vi cc bn cha bit ti h thc lng trong tam gic th c th lm theo cchkdyDE, CFsongsongviAC, BDtngngripdngnhlPtolemychocctgicnitipABCE, ACDFsuyraktqu.Bi2.78.pdngnhlhmssinvbtngthcquenthuccos A + cos B + cos C 32.38III.Ligiichitit1.Ccbitonntptuynsinhlp10Bi1.1 TamgicABCvungti AcBC=2AB. LyD, EnmtrnAC, ABsaocho

ABD=13

ABCv

ACE=13

ACB.FlgiaoimcaBD, CE.H, KlimixngcaFquaAC, BC.(a) ChngminhH, D, Kthnghng.(b) ChngminhtamgicDEFcn.Li giiIHKVTFACBDE(a)GiT= FH AC, V= FK BC.TgiithitcthsuyratamgicABClnatamgicunnvictnhccgcltmthng.Tac,

FHD =

HFD =

ABD = 20.Mckhc,

FHK=

FTV (doTV | HK)=

ACE(doCTFV nitip)= 20=

FHDSuyraH, F, Kthnghng.(b)HKctBCtiI.Talnlttnhccgc:

DFI= 180

DIF

IDF= 1802040= 120

BEC= 90 + 10= 100v

BIF= 800nnBEFInitip.Suyra___

EFI= 180

ABC= 120=

DFI

FIE= 20=

DIFDo, DFI= EFI FD = FE.Do,tamgicDEFcntiF. Bi1.2 ngtrn(O)nitiptamgicABC(AB> AC)tipxcviAB, ACtiP, Q.Gi R, SlnltltrungimBC, AC. GiaoimcaPQ, RSlK. ChngminhrngB, O, Kthnghng.Li gii39KRSQPOABCTrctin, taschngminhrngRB=RK. Gi a=BC, b=CA, c=AB, chrngSK= SQdotamgicSQKc2gcybngnhau.Khi:RK= RS SK=c2 SQ =c2 (CS CQ)=c2 _12 b a +b c2_=c2 12 b +a +b c2=12 a = BRVvy,tamgicBRKcntiR,suyra

RBK=

RKB=

KBA(RK | AB).DoKthucngphngicgc

ABChayB, O, Kthnghng. Bi1.3 ChotamgicABCnhnnhnHlmtrctm.Chngminhrng,tacbtngthc:HA +HB +HC 0vEF> 0,khaicnhaiv,tacMA +MB +MCEFyliucnchngminh. Bi2.9 ChotamgicABCcBC=a, CA=b, AB=cvIltmngtrnnitiptamgicABC.ChngminhrngIA +IB +IC ab +bc +caLi giicbaDIAB C83Tassdngb:IA =_bc(b +c a)a +b +cChngminhb.GiDlchnngphngictnhA.Theocngthcngphngic:IAc=IDBD=ADc +BDTccngthcBD =acb +cAD2=4bc(b +c)2p(p a)Tasuyra:IA =_bc(b +c a)a +b +cBcchngminh. Theob,tacnchngminhrng:_bc(b +c a) +_ca(c +a b) +_ab(a +b c) _(a +b +c)(ab +bc +ca)Bnhphnghaiv,tac

(b2c +bc2) 3abc + 2_abc2[c2(a b)2]

(b2c +bc2) + 3abcBtngthctrntngngvi

_(b +c a)(c +a b)ab 3nytacthsdngAM-GMnhsau:

_(b +c a)(c +a b)ab=

_b +c ab

_c +a ba

12

_b +c ab+c +a ba_= 3BtngthccuicchngminhnnsuyraIA +IB +IC ab +bc +caChngminhhontttiy. Bi2.10 TimAnmngoi ngtrn(O), khai tiptuynAB, ACn(O). GiE, FltrungimcaAB, AC.LyDlmtimbtktrnEF,vcctipDP, DQtingtrn.PQctBC, EFlnlttiN, M.Chngminhrng,ON | AM.Li gii(i)Cch1.84KSLTNMPQFEAOBCDXt cc - i cc i vi ng trn (O, R) : Al cc ca BC, D l cc ca PQm BCPQ = NnnNchnhlcccaADiviO.AD ON.Mckhc,tED OAsuyra,DO2DA2= EO2EA2=12_OA2+OB2_14AB214AB2= R2VvyDA2= DO2R2= TD/(O)ng thc ny chng t Dl tm ca ng trn ngoi tip tam gic APQ. Xt cc - i ccivingtrnny:OlcccaPQnnM, Olinhp.Hnna,DM OAnnMlcccaOA.DoAM AD.TysuyraON | AM(iucnchngminh).(ii)Cch2.Theochngminhcch1,taccDA = DP= DQ = r.Hnna,doE, FltrungimAB, ACnnEFchnhltrcngphngca(O; R)v(A; 0). TsuyraMAltiptuyn(APQ)hayAM DA.ODctBC, PQT, LvOActBC, PQK, S. TacOLOD=R2=OKOAnnAKLDnitip.DthyrngSKTLcngnitipnnAD | ST.TamgicSTOnhnNlmtrctmnnST ON.Do,AD ON.TysuyraON | AM. 85Bi2.11 ChotamgicABCcnti Ani tipngtrn(O). TrncnhyBC, lyimM(MkhcB, C).VngtrntmDquaMtipxcviABtiBvngtrntmEquaMtipxcviACtiC.GiNlgiaoimthhaicahaingtrnny.(a) Chngminhrngtngbnknhcahai ngtrn(D), (E)lkhngi khi MdingtrnBC.(b) TmtphptrungimIcaDE.Li giiFKHINEDPQAOB CM(a)Gi KlgiaoimcaBD, CE. ChrngcctamgicDBM, EMC, BKCcnnnDM | CK, EM | BKvBK= CK= kkhngi.pdngnhlThales,tacDMCK=BMBC , EMBK=CMBCSuyraR(D) +R(E)k=BM+CMBC= 1VvyR(D) +R(E)= kkhngi.(b)GiP, Q, H, FlnltlhnhchiucaD, E, I, KlnBC.DPKF+EQKF=BDBK+CECK= 1SuyraDP+ EQ = KF= khngi.TyIH=2cngkhngi.Do,IdichuyntrnngthngsongsongvcchBCmtkhong2khngi. Bi2.12 Cho Ml im di ng trn ng trn (O, r) c hai ng knh c nh AB, CDvunggcvinhau.GiIlhnhchiucaMlnCDvPlgiaoimcaOM, AI.TmtphpccimP.Li gii86yxPIBCA ODMChn h trc ta nhn O lm gc v A(r, 0), B(r, 0), C(0, r), D(0, r) v M(r cos , r sin ).Khitac:PhngtrnhngthngCD : x = 0.PhngtrnhngthngIM: y= r sin .TsuyrataimIlI(0, r sin ).PhngtrnhngthngOM:xr cos =yr sin PhngtrnhngthngAI:x +rr=yr sin SuyraPctathamnhphngtrnh:___xr cos =yr sin x +rr=yr sin ___tan =yxsin =yx +rTalictan2 =sin21 sin2,suyra:y2x2=y2(x +r)21 y2(x +r)2ngthcnytngngviy2= 2xr +r2.VytphpccimPlparabolcphngtrnhy2= 2xr +r2. 87Bi2.13 ChotamgicuABCvmtimMbtk trongmtphngtamgic. Gix, y, zl khong cch t Mn cc nh A, B, Cv p, q, rl khong cch t Mn cc cnhAB, BC, CA.Chngminhrng:p2+q2+r2

14(x2+y2+z2)Li giiC'B'A'BACMNuMtrngvi mttrongccnhA, B, Cth dthybtngthccnchngminhlng.Xt trng hp Mkhng trng vi nh no ca tam gic ABC. Gi A

, B

, C

ln lt l hnhchiuvunggccaMlnccngthngBC, CA, ABtheothtvGltrngtmtamgicA

B

C

.TheonhlLeibniz,tacMA2+MB2+MC2= 3MG2+13_B

C2+C

A2+A

B2_

13_B

C2+C

A2+A

B2_Mt khc, tamgicAB

C

ni tipngtrnngknhAM, do

B

AC

=60hoc

B

AC

= 120.Vvy(theonhlsin)B

C

= MAsin 60 (= MAsin 120) =x32SuyraB

C2=3x24.Tngt,tacC

A2=3y24, A

B2=3z24.DoB

C2+C

A2+A

B2=34_x2+y2+z2_Vvyp2+q2+r2

14_x2+y2+z2_ychnhlbtngthccnchngminh. 88Bi2.14 ChoagicuA1A2A3A4A5A6A7vimMbtk trongmtphng. ChngminhrngMA1 +MA3 +MA5 +M7MA2 +MA4 +MA6Li giiA1OMA2A3A4A5A6A7tA1A2= a, A1A3= b, A1A4= c.pdngnhlPtolemy:ivitgicA1A2A3M:a(MA1 +MA3)bMA2(1)ivitgicA5A6A7M:a(MA5 +MA7)bMA6(2)ivitgicA2A4A6M:b(MA2 +MA6)cMA4(3)T(1)v(2)suyra:a(MA1 +MA3 +MA5 +MA7)b(MA2 +MA6) (4)T(3)v(4)suyra:a(MA1 +MA3 +MA5 +MA7)cMA4(5)T(4)v(5)suyra:a(MA1 +MA3 +MA5 +MA7)_1b+1c_ MA2 +MA4 +MA6(6)pdngnhlPtolemychotgicnitipA1A3A4A5,tac:ab +ac = bc a_1b+1c_= 189Thayvo(6)tac:MA1 +MA3 +MA5 +MA7MA2 +MA4 +MA6Taciucnchngminh. Bi2.15 Tamgic ABC khng cn ni tip (O) c A1, B1, C1l trung imcaBC, CA, AB. Gi A2lmtimtrntiaOA1saocho2tamgicOAA1vOA2Angdng.CcimB2, C2nhnghatngt.ChngminhrngAA2, BB2, CC2ngquy.Li giiVTA2B2C2A1B1C1OABC(i)Cch1.ThaitamgicOAA1vOA2AngdngsuyraOA1OA2= OA2= R2.Do,A2chnhlgiaoimcctiptuyntiB, Cca(O).ng thng qua A2song song vi tip tuyn ca (O) ti A ct AB, ACti T, V . Do

A2BT=

A2TBnnA2B=A2T. Mtcchtngt, A2T=A2B=A2C=A2V . V th, BCV Tnitip(A2)hay ABCAV T.LicA1, A2lnltltrungimBC, TV nn AA1CAA2T.Suyra

CAA1=

TAA2.ngthcnychngtAA2lngitrungcatamgicABC.Do,ccngthngAA2, BB2, CC2sngquytiimLemoinecatamgicABC.(ii)Cch2.Theochngminhcch1th (O) chnhlngtrnni tipcatamgicA2B2C2. DoBA2= CA2, CB2= AB2, BC2= AC2nn:CA2CB2

AB2AC2

BC2BA2= 190TheonhlCeva,tacngayAA2, BB2, CC2ngquy. Bi2.16 ChotamgicABCviMltrungimBC.Vngtrn(O)tyquaAvctcconAB, AC, AMlnlttiB1, C1, M1.ChngminhrngAB1 AB +AC1 AC= 2AM1 AMLi giiC1M1B1MAB COTac___AB1 AB= AB2BB1 AB= AB2TB/(O)AC1 AC= AC2CC1 AC= AC2TC/(O)2AM1 AM= 2AM22TM/(O)= AB2+AC2BC222TM/(O)Do,chcnkimtrangthcsaul:TB/(O) +TC/(O)2TM/(O)=BC22ngthcnytngngvi:OB2+OC22OM2=BC22(ngtheocngthctrungtuynchotamgicOBC).Vvy,bitoncchngminhhontt. Bi2.17 ChotamgicABCni tipngtrnbnknhR.Gi qlchuvi tamgiccccnhltmccngtrnbngtiptamgicABC.Chngminhrng:q63RLi gii91OIbIIaIcABCNidungcabitonthcchtlskthptrctipcahaibsau:B1:ChotamgicXY Znitipngtrn(O, R).KhiXY+Y Z +ZX33R.Chngminh.GiGltrngtmtamgicXY Z,khitheonhlLeibniz,tac9R2(XY2+Y Z2+ZX2) = 9OG2 0KthpvibtngthcCauchy-Schwarz,tac27R2 3(XY2+Y Z2+ZX2) (XY+Y Z +ZX)2Tngngvi33RXY+Y Z +ZXB1cchngminh. B2:ChotamgicABCni tip(O, R). Ia, Ib, IctheothtltmngtrnbngtipccgcA, B, C.KhingtrnngoitiptamgicIaIbIccbnknhbng2R.Chngminh.VAIavIbIclccngphngictrongvngoicagc

BACnnIaAIbIc.Do A, B, Cl chn cc ng cao trong tam gic ABCnn (ABC) l ng trn Euler catamgicIaIbIc.Vvybnknhngtrn(IaIbIc)bng2R.B2cchngminh. Bi2.18 ChotamgicABCc:BC=a; CA=b; AB=c;vrvRtheothtlbnknhngtrnnitipvngoitiptamgicABC.ChngminhrngrR+(a b)2+ (b c)2+ (c a)216R2

12Li gii92M DIOAB CGi Il tm ng trn ni tip tam gic ABCv Dl tip im ca ng trn ni tip (I)trncnhBC,MltrungimcaBC.Khngmttnhtngqut,gisbac.Khi:R22Rr = OI2 DM2=(b c)24TngngvirR+(b c)28R2

12Mtkhc:(b c)2= (a b)2+ (c a)2+ 2(a b)(c a) (a b)2+ (c a)2Suyra:rR+(a b)2+ (b c)2+ (c a)216R2

12Chngminhhontt. Bi2.19 Cho tam gic ABC. Cc ng phn gic BE, CFct nhau ti I. AIct EFtiM. ngthngquaMsongsongvi BCtheothtctAB, ACti N, P. ChngminhrngMB +MC< 3NPLi gii93S R HKLPFMENIOAB Cutin,tachngminhbsauy:B : Chotamgic ABC, cphngic BD, CE. LyimMbt k thuc DE. KMH BC, MK AC, ML AB.KhitacMH= ML +MK.Chngminhb.GiTlgiaoimDFvMH.TE, DvEF, DO BC; DN AB; EP AC.Suyra:EF= EP; DN= DO.TheonhlThales,tacMKEP=MDDE=MTEFDoEF= EPnnMT= MK (1)CngtheonhlThales,tacMLDN=EMED=FHFO=HTDOMDO = DNnnTH= ML (2)T(1), (2)suyraMT+TH= MH= ML +MKBcchngminh. Trlivibiton.GiH, K, LtheothtlhnhchiucaMlnBC, CA, ABQuaMkMR | ABvMS | AC.pdngbtacMH= ML +MK= 2ML = 2MKChrng: MRHMNLv MSHMPK.SuyraMR = 2MNvMS= 2MP.pdngbtngthctamgic,tacMB +MC< (MR +BR) + (MS +SC)= 3(MN+MP)= 3NP94Taciucnchngminh. Bi2.20 ChotamgicABCnhnvingcaoCFvCB>CA.GiO, Hlnltltmngoi tipvtrctmcatamgicABC. ngthngquaFvunggcvi OFctACtiP.Chngminhrng

FHP=

BAC.Li giiQMNPHKFOABCGi Kl im i xng ca Hqua AB, khi K (O). ng thng PFct (O) v BK, AClnlttiM, M, Q, P,trongP, NthuccngmtnamtphngbCKkhngchaB.XtdycungMNcOF MNnnFltrungimcaMN.Do,pdngnhlconbmchodycungMN, tathyrngFcngltrungimcaPQ. Mckhc, FltrungimHKnnPHQKlhnhbnhhnh.Vy

PHF=

BKC=

BAC,taciucnchngminh. Bi2.21 Chongtrn(O; R)vmtimPcnhbntrongngtrn.AB, CDl2dycungdingca(O)nhngluniquaPvlunvunggcvinhau.(a) ChngminhrngPA2+PB2+PC2+PD2khngi.(b) GiIltrungimBC.HiIdingtrnngno?Li giiMIKABOCDP95(a)pdngnhlPythagore,tathyrng:PA2+PB2+PC2+PD2= AC2+BD2= BC2+AD2VngknhAKcangtrn(O). Khi , BK ABmAB CDnnBK |CD.HnhthangBCDKnitipnnlhnhthangcn.TysuyraCK= BD.pdngnhlPythagorechotamgicACKvungtiC: AC2+CK2= AK2SuyraAC2+BD2= 4R2hayPA2+PB2+PC2+PD2= 4R2khngi.(b)Trctin,taschngminhrng:IO2+IP2= R2Thtvy,pdngnhlPythagorechotamgicOIBvungtiI,tathuc:OB2= OI2+IB2TamgicPBCvungtiPcIltrungimBCnnPI= IB.Do:R2= OI2+IP2Gi Ml trung im OP. Theo cng thc ng trung tuyn (c th chng minh da vo kinthclp9):IM2=2(IP2+IO2) OP24=2R2OP24DoIdichuyntrn_M;2R2OP24_cnh. Bi2.22 Cho tam gic ABCv im Mbt k nm trong tam gic . Chng minh rng :MA +MB +MC + minMA, MB, MC < AB +BC +CALi giiTrcht,tachngminhbsauy:B:ChotgicABCDvimMbtknmtrongtgic.Chngminhrng:MD +MC< DA +AB +BCChngminhb.LABCDM96XtMnmtrongtamgicDBC.GiLlgiaoimcaDMvBC.pdngbtngthctamgic,tacDA +AB +BCDB +BC= DB +BL +LC DL +LC= DM+ML +LC DM+MCTngtxtMnmtrongtamgicABD,tachngminhc:AD +AB +BCDM+MCSuyraiucnchngminh. Trlivibiton.DFEAB CMGiD, E, FtheothtltrungimcaBC, CA, AB.Dthyvi mi imMthuctamgicABCth tnti tnhthai trongbahnhthangBCEF, CAFD, ABDEchan.Khngmttnhtngqut,gisMnmtronghnhthangBCEFvABDE.pdngb,tac___MA +MB 2HK.Do,taluncP(MAB) > 2HK.ngthcxyrakhivchkhiOltmbngtipgcMca MAB. B 2 : Cho tam gic ABC, Ml im nm trong tam gic. AM, BM, CMct BC, CA, ABtheothtD, E, F.TacAMAD+BMBE+CMCF= 2Vicchngminhb2khngin,xindnhchobnc. Trlibiton:Qua Mk ng thng song song vi EF, ct AB, ACti X, Y ; song song vi FD ct BC, BAtiZ, T;songsongviDEctCA, CBtiU, VCctamgicMUT, V MX, ZY MngdngvitamgicDEFtheocctstngngl:AMAD , BMBE , CMCFTb1,tacP(MUT) +P(V MX) +P(ZY M)2IK + 2KH + 2HI.TsuyraP(MUT) +P(V MX) +P(ZY M)2P(HIK). (1)Tb2,tac:AMAD+BMBE+CMCF= 2Suyra2P(DEF) =AMAD P(DEF) +BMBE P(DEF) +CMCFP(DEF)Tngngvi2P(DEF) = P(MUT) +P(V MX) +P(ZY M) (2)T(1)v(2)tasuyraiucnchngminh. 120Bi2.47 TamgicABCnhnnitip(O),ngcaoAHct(O)tiA

.OA

ctBCtiA

.XcnhtngtchoB

, C

.ChngminhAA

, BB

, CC

ngquy.Li giiB''C''A''C'B'A'OABCTac

OBC= 90 Av

CBA

=

CAA

= 90 C,suyra

OBA

= B.LicOB= OA

nn

OA

B= B.Suyra

BOA

= 1802B.Tngttac

COA

= 1802C.pdngnhlsintrongtamgictacBA

sin

BOA

=OA

sin

OBA

vCA

sin

COA

=OA

sin

OCA

SuyraBA

sin

BOA

=CA

sin

COA

DoBA

CA

=sin(1802B)sin(1802C)=sin 2Bsin 2CTngtchohaiimE, FvpdngnhlCeva,taciucnchngminh. Bi2.48 Chongtrn(O)vmtngthngdcnh.GiHlhnhchiucacaOtrnd. LyMcnhthucngtrn. A, Bthayi trndsaochoHltrungimAB.GisAM, BMct(O)tiP, Q.ChngminhPQluniquamtimcnh.Li gii121RNKQTSPB HOMANu M, O, Hthng hng, khi ta c PQ lun song song vi (d). Do ta ch xt trng hpM, O, Hkhngthnghng.Gis(d)khngct(O)(cctrnghpkhcchngminhtngt).Khng mt tnh tng qut, gi s Mv Bcng pha so vi OH. T Pk ng thng d

songsongvidctMH, MBtngngtiS, T.GiNltrungimcaPQvRlgiaoimkhcMcaMHvi(O).TacNS | QT,suyra(NP, NS) = (QP, QT) = (RP, RS) (mod)DoP, N, R, Sngvin.Vvy(RN, RH) = (PN, PS) = (KN, KH) (mod)hayN, R, H, Kngvin,m

ONK=

OHK=2nnO, H, N, Kngvin.NhvytasuyraKlgiaoimca(d)vi(OHR)nnKlimcnh.VyPQluniquaKcnh. Bi2.49Cho ng trn tm I ni tip tam gic ABC tip xc vi BC, AB, AC ti D, E, F.QuaEvngsongsongviBCctAD, DFM, N.ChngminhrngMltrungimcaEN.Li giiPNMEDFIAB C122QuaAdngngthng(d)songsongviBCvctDFtiP.TcchdngtrnsuyraMN | AP.DotheonhlThalestacMNAP=DMADMtkhc,cngtheonhlThales,tacEMAE=CDCA=CECA=DMADThaingthctrntasuyraEMAE=MNAPMtkhc,dthyAP= AF= AEnnsuyraEM= MN.VyMltrungimEN. Bi2.50 ChotamgicABCcAB=c, BC=a, AC=bvI ltmngtrrnnitip. Hai im B

, C

ln lt nm trn hai cnh AB, ACsao cho B

, C

, Ithng hng. ChngminhrngSABC a +b +c2bc

_SAB

CSABC

Li gii(i)Cch1.C'DIAB CB'GiDlchnngphngictronggcA.Trctintacccktququenthucsau:BD =acb +cAD =bb +cAB +cb +cACVIlchnngphngictrongcatamgicABDnn:AIDI=BABD=b +caAIAD=b +ca +b +c123Tasuyra:AI=b +ca +b +cAD=b +ca +b +c_bb +cAB +cb +cAC_=ba +b +cAB +ca +b +cAC=bAB(a +b +c)AB

AB

+cAC(a +b +c)AC

AC

MtkhcB

, I, C

thnghngnnbABAB

(a +b +c)+cACAC

(a +b +c)= 1.Tngngvi:a +b +c =bABAB

+cACAC

AMGM 2 _bABAB

cACAC

= 2bc _ABACAC

AC ABACAC

AB= 2bc S2ABCSAB

CSAC

BSuyra:SABC a +b +c2bc

_SAB

CSABC

nychngminhhontt.(ii)Cch2.Bnhphngvchuynv,btngthcubitngngvi:4bc(a +b +c)2 SAB

CSABC

SABC

SABC4bc(a +b +c)2 AB

AC

ABAC4b2c2(a +b +c)2AB

AC

Tacbsau:AB

AC

IA2cos2 A2Xinkhngchngminhbny,bnccthxemnhbitp.Tiptheo,tacccngthc:IA =_bc(b +c a)a +b +c,124cos2 A2=cos A + 12=b2+c2a22bc+ 12=(b +c a)(a +b +c)4bcTthyrng:AB

AC

IA2cos2 A2=4b2c2(a +b +c)2Btngthccchngminh. Bi2.51 ChotgicABCDni tip. E, F, G, Hlnltltmngtrnni tipcctamgicABC,BCD, CDA, DAB.ChngminhrngtgicEFGHnitip.Li giiHG FEABDCTac

DGC= 90 +

DAC2= 90 +DBC2=

DFCSuyratgicDGCFnitip.Tngt,cctgicCFEB, AHEB, AHGDnitip.Tsuyra

EFG = 360_

EFB +

BFC +

CFD +

DFG_= 360_

ECB +

BFC +

CFD +

DCG_= 360_

ACB2+ 90 +

BAC2+ 90 +

DAC2+

ACD_= 90125Chngminhtngtchoccgccnli,tasuyraEFGHlhnhchnht. Bi2.52 ChohnhvungABCD.ItythucAB, DIctBCtiE,CIctAEtiF.ChngminhrngBF DE.Li giiGKTFED CA BIChoBF, AClnltctDEtiT, K.Suyra(KTIE) = 1.GigiaoimcangtrnngoitipABCDviDElN.ANctBCtiG.Tac:

DNC=

CNB=

BNG =

CNE= 45.SuyraNClphngicngoivNGlphngictrongcatamgicBNE.Do(CGBE) = 1hay(KNIE) = 1(xtphpchiuxuyntmA)VvyN T.Taciucnchngminh. Bi2.53 Cho tam gic ABCkhng vung ni tip ng trn (O), trc tm H. d l ngthng bt k qua H. Gi da,db, dc ln lt l cc ng thng i xng vi d qua BC, CA, AB.Chngminhrngda,db,dcngquytimtimtrn(O).Li giiMH3HABC126GiH1, H2, H3lnltlccgiaoimthhaicaAH, BH, CHvi(O).TacSAB: H H3.ChonnH3 dc.TngtH1 da, H2 db.MtkhcSAB: dcdvSBC: d da.DoSBC SAB= R[B,2(BA,BC)]: dc daSuyra(dc, da) 2(BA, BC) (mod)GigiaoimcadavdclM.Tac:(CH3, CH1) 2(CH, CB) 2_2 (BA, BC)_(mod)NhvythMCH3H1nitipsuyraMnmtrn(ABC).Mtkhc,tac(dc, MH2) (CH3, CH2) 2(AB, AC) (mod)NhngdbliquaH2vtovidcmtgc2(AB, AC)(chngminhtngttrn).Nhvy,MH2trngvidb,taciucnchngminh. Bi2.54 ChohnhthangABCD(AB |CD). ACctCDti O. BitkhongcchtOnADvBCbngnhau,hychngminhrngABCDlhnhthangcn.Li giiOHNDMABCGi Hl giao imca AD, BC. M, Nlnlt l giao imca cc cpng thng(HO, AB), (HO, CD).SuyraM, NlnltltrungimcaAB, CD.V cc khong cch t On AD, CBbng nhau nn HOl phn gic

DHC. Suy ra tam gicHDCcntiHdocngphngiccnglngtrungtuyn.VyABCDlhnhthangcn. Bi2.55 Cho tam gic ABCcn ti A. ng trn tip xc AB, AC, ct BCti K. AKct ti im th hai l M. P, Q l im i xng ca Kqua B, C. Chng minh rng ngtrnngoitiptamgicMPQtipxcvi.Li gii127EMQ PKDC BAGiD, ElnltltipimcaviAB, BC; P

lgiaoimcaMDvBC.Ta c DMEKl t gic iu ha nn D(DEKM) = 1 hay D(BEKP

) = 1. M DE | P

KnnBltrungimP

KhayP

P.VvymM, D, Pthnghng.TngttacngcM, E, Qthnghng.LicDE | PQnntntimtphpvt :binDEthnhPQ.Suyra :: (MDE) (MPQ).Vyhaingtrnv(MPQ)tipxcvinhautiM. Bi2.56 ChotamgicABCvungti Ac B=20, phngictrongBI. imHnmtrncnhABsaocho

ACH= 30.Hytnhso

CHI.Li gii(i)Cch1.LKHIABCKphngicCKcagc

HCB.GiLlhnhchiucaKtrnBC.Hai tamgic BLKv BAClnltvungti L, AvcgcBchungnnchngngdng,suyraLBAB=KBBC=KHCH128LictamgicBKCcntiKnnLltrungimBC.VvymBCAB=2KHCH=KHAHHayICIA=HKHAT,theonhlThalesthtacHI | CK.Vy

CHI=

HCK= 20.(ii)Cch2.t

CHI=

AHI= 60.pdngnhlsinchotamgicCHItacCIsin

CHI=HIsin

ACHSuyraCIsin =HI12(1)Talic:HIAI=1sin

AHI=1sin(60)(2)T(1)v(2)tac:CIAI=CIHI HIAI=2 sin sin(60)=2 sin cos(30 +)MBIlphngic

ABCnnAICI= cos 20.Suyracos(30 +)2 sin = cos 20haycos(30 +) = 2 cos 20 sin ()M060nnvtrilhmnghchbin,vphilhmngbinDophngtrnh()cnghimduynht = 20.Vy

CHI= 20. Bi2.57 ChotamgicABCngoi tip(I). Gi D, E, Flnltlimi xngvi IquaBC, CA, AB.ChngminhrngAD, BE, CFngquy.Li gii129EDFIAB CTac:sin

BADsin

CAD=sin

BADsin

ABD

sin

ACDsin

CAD

sin

ABDsin

ACD=BDAD ADCD sin 3B2sin 3C2=IBIC sin 3B2sin 3C2Chngminhtngtchosin

ACFsin

BCFvsin

CBEsin

ABE,tasuyra:

sin

BADsin

CAD=

IBIC

sin 3B2sin 3C2= 1TheodngnhlCevadngsin,taciucnchngminh. Bi2.58 ChotamgicABCcntiAnitip(O).imMltrungimcaAC.BMctli(O)tiimthhailQ.Chngminhrng2AQBQ.Li giiQMAOB C130tAB= AC= a, BC= b (2a > b).Tac:2BM=a2+ 2b2TamgicAMQngdngvitamgicBMCnn:AQBC=AMBM AQ =BCAMBM=aba2+ 2b2Theohthclngtrongngtrnth:MQ MB= MAMC MQ =a22a2+ 2b2Vytacnchngminh4aba2+ 2b2

a2+ 2b2+a2a2+ 2b2Tngngvi4aba2+ 2b2+a2Hay(a b)2 0.Btngthcnyhinnhinngnntaciucnchngminh. Nhnxt.Mtktqurnghnhnl:BQmaxAC, 2AQChaibtngthcucchngminhtngthcBQ =12a_bAQ+AQb_Trong,a = AB= AC, b = BC.Vicchngminhngthcnyxindnhchobnc.Bi2.59 Cho ABCtha mn AB+BC= 3CA. ng trn ni tip (I) tip xc AB, BCtiD, E.GiK, LtngngixngviD, EquaI.ChngminhrngtgicACKLnitip.Li giiGFMLKEDIBCA131GiGlgiaoimCK, AB;FlgiaoimAL, BC;MlgiaoimAL, CK.tBC= a, CA = b, AB= c,a +b +c2= p.DthyBG = AD = p a.DoAG = c (p a) = c +a p =c +a b2= b(doa +c = 3b)Suyra AGCcntiA.Tngt,tac ACFcntiC.Ttac

KML =

AGC +

BAF= 90

BAC2+

BAC 90 +

BAC2=

BAC +

ACB2=12_180

ABC_=12

KILSuyraM (I).Do

MLK=

MDK=

DGK=

ACGVytaciucnchngminh. Bi2.60 ChotamgicABCngoitip(I).(I)tipxcBC, CA, ABlnlttiD, E, F.ChngminhrngtmngtrnngoitipcctamgicAID, BIE, CIHthnghng.Li giiAC1B1A1DEFIBCGiA1, B1, C1lnltltrungimcaEF, FD, DE.DoDA1, EB1, FC1ngquytitrngtmGcatamgicDEF.XtphpnghchotmIphngtchk= r2,binDA1, EB1, FC1thnhngtrnngoitipcctamgicIAD, IBE, ICF.MDA1, EB1, FC1ngquynnccngtrncngcngiquamtimkhcI.Ttaciucnchngminh. 132Bi2.61 Cho tam gic ABCni tip (O). M, Nln lt l im chnh gia cung ABkhngchaCvcungACkhngchaB. DltrungimMN. Glmtimbtk trncungBCkhngchaA.GiI, J, KlnltltmnitipcctamgicABC, ABG, ACG.LyPlgiaoimthhaica(GJK)vi(ABC).ChngminhrngP DI.Li giiDoG, J, MvG, K, NthnghngvtgicPJKGni tipnn

PJM=

PKN. Li c

PMJ=

PNKnn PJMPKN,suyraPMPN=JMKN.MJM= AM,KN= ANnnPMPN=AMANhaytgicAMPNiuha.Gi P

lgiaoimDI v(O) (P

thuccungBCkhngchaA). TacMA=MI vNA = NInnAixngviIquaMN.VvyMNlngphngiccagctobihaingthngDA, DP

.SuyratgicAMP

Niuha.DoP P

vtaciucnchngminh. Bi2.62 ChongicuA1A2. . . An(n4)thamniukin1A1A2=1A1A3+1A1A4Hytmn.Li giiGiRlbnknhngtrnngoitipagic.pdngnhlsintacA1A2= 2Rsin n, A1A3= 2Rsin 2n, A1A4= 2Rsin 3n133tn= x_0 < x 4_.Tacdyccngthctngngsau1sin x=1sin 2x+1sin 3x1sin x=sin 2x + sin 3x2 sin 2x sin 3x1sin x=sin 2x + sin 3x2 sin x cos x sin 3xsin 2x + sin 3x = 2 sin 3x cos xsin 2x + sin 3x = sin 2x + sin 4xsin 3x = sin 4xM0 < x 4nnx =7.Do,n = 7lgitrduynhtcntm. Bi2.63 GiAA1, BB1, CC1tngnglccngphngictrongcatamgicABC.AA1, BB1, CC1ct ng trn ngoi tip tam gic ti A2, B2, C2theo th t. Chng minhrng:AA1AA2+BB1BB2+CC1CC2

94Li giiA2A1B2B1C2C1OAB Cpdnghthclngtrongngtrn,tacAA1 A1A2= A1BA1CMcchthcquenthucchota:A1B=acb +cA1C=abb +cAA1= la=_bc(a +b +c)(b +c a)b +c134Suyra:A1A2=a2bc(b +c)_(a +b +c)(b +c a)Khi:AA1AA2=AA1AA1 +A1A2=(a +b +c)(b +c a)(b +c)2= 1 a2(b +c)2Dovybtngthccnchngminhtngngvi:

a2(b +c)2 34TheobtngthcCauchy-SchwarzvNesbitt,tac:3

a2(b +c)2 _

ab +c_2

94Tytasuyraiucnchngminh. Bi2.64 Cho tam gic ABC, ng thng d ct cc ng thng BC, CA, ABln lt tiD, E, F. Gi O1, O2, O3ln lt l tm ng trn ngoi tip cc tam gic AEF, BDF, CDE.ChngminhrngtrctmtamgicO1O2O3nmtrnd.Li giiMO3O2O1EABCDFGiMlimMiquelcatgictonphnBCEFAD.TalicimixngcaMquaO1O2, O2O3, O3O1lnltlD, E, F.SuyraMthucngtrnngoi tiptamgicO1O2O3vdlngthngSteinercaMivi(O1O2O3).SuyratrctmcatamgicO1O2O3nmtrnd.Taciucnchngminh. 135Bi2.65 Cho t gic ABCD, ACct BDti O. Gi M, N, P, Q ln lt l hnh chiu caOtrn AB, BC, CD, DA. Bit rng OM= OP, ON= OQ. Chng minh rng ABCDl hnhbnhhnh.Li giiQPNMOEFADBCTrnghpAB | CDhocBC | ADthhinnhintaciucnchngminh.Trnghpkhngsongsongtaschngminhbngphnchng.GisABCDkhngphilhnhthang.GiElgiaoimcaAD, BC;FlgiaoimcaAB, CD.TgithittacEO, FOlnltlphngictrong

AEB,

CFB.TalicE(ABOF) = 1nnEFlphngicngoica

AEB,suyraEF EO.TngttacEF FO.DoFO | EO(vl)Tsuyraiucnchngminh. Bi2.66 ChotamgicABC,phngictrongAD(D BC).GiM, NlccimthuctiaAB, ACsaocho

MDA=

ABC,

NDA=

ACB.CcngthngAD, MNctnhautiP.Chngminhrng:AD3= ABACAPLi giiNMDAB C136Tac:

MDN=

MDA +

NDA = B + C= 180 A.DotgicADMNnitip.Suyra

AMP=

ADN= Cv

ANP=

ADM= B.Vvy AMPACDv ANPABD.TtacccngthcAMAD = APAC (1)ANAD = APAB (2)Mtkhc, ADNACD.SuyraAD2= ANAC (3)Tngttac:AD2= AMAB (4)Nhnvtheovccngthctrntac:AMANAD6= AMANAB2 AC2 AP2TngngviAD3= ABACAPBitoncchngminh. Bi2.67 Trn mt phng cho 2000 ng thng phn bit, i mt ct nhau.Chng minhrngtntitnht2ngthngmgccachngkhnglnhn1802000().Li giiXt mtimObt k,qua ta v2000ng thng tng ng song song viccng cho.Khiccgcgia2ngthngboton.2000ngthngtrntothnh4000tiachunggcO. Mi cptialintiptngngvimtgcgia2ngthngnncng4000gcv4000gcctngsol360.TheonguynlDirichlettaciucnchngminh. Bi2.68 ChotgicABCDnitip(O)cAB= AD.M, NnmtrncccnhBC, CDsaochoMN= BM+DN.AM, ANct(O)tiP, Q.ChngminhrngtrctmtamgicAPQnmtrnMN.Li gii137JCN'DBNHAP QMGiHl im trnonMNsao choMH= BM, NH= DN; N

l im trntia ica tiaBCsaochoBN

= DN.Tac

ABN

=

ADNvAB= ADnn ABN

= ADN.SuyraAN

= AN.LicMN

= MB + BN

= MB + DN= MN.DoN

vNixngvinhauquaAP.TsuyraHixngviBquaAP.Tngt,tacHixngviDquaAQ.Gi JlgiaoimcaAHvi (O)th P, QltrungimcacccungBJ, DJkhngchaA. Suy ra PJ= PB= PH, QJ= QD = QHhay Hi xng vi Jqua PQ. V vy AHPQ.Suyra

PQH=

PQJ=

PAJ=

PAB=

PQBDoB, Q, HthnghnghayQHAP.VyHltrctmtamgicAPQ Bi2.69 ChotgicABCD.HaingchoAC, BDctnhautiO.Gir1, r2, r3, r4lnltlbnknhccngtrnnitipcctamgicAOB, BOC, COD, DOA.Chngminhrng1r1+1r3=1r2+1r4liukincnvtgicABCDngoitipcmtngtrn.Li giitAB= a, BC= b, CD = c, DA = dvOA = x, OB= y, OC= z, OD = t.Gi =

AOB.Khitacdyccngthctngngsau:1r1+1r3=1r2+1r4pAOBSAOB+pCODSCOD=pBOCSBOC+pAODSAODx +y +axy sin +z +t +czt sin =y +z +byz sin +x +t +dxt sin axy+czt=byz+dxt138azt +cxy= btx +dyza2z2t2+c2y2x2+ 2acxyzt = b2x2t2+d2y2z2+ 2bdxyzt2zt cos 2xy cos + 2ca = 2xt cos + 2yz cos + 2bd(c2z2t2) + (a2x2y2) + 2ca = (d2x2t2) + (b2y2z2) + 2bd(a +c)2= (b +d)2a +c = b +dng thc cui cng chng t t gic ABCD ngoi tip v cc ng thc trn u tng ngvinhaunntaciucnchngminh. Bi2.70 Cho tam gic ABCc Ml trung im ca BCv Hl trc tm tam gic. ngthng vung gc vi HMti Hct AB, ACti D, E. Chng minh rng Hl trung im caDE.Li giiDEHMABCPQ(i)Cch1.Tac:

DAH=

MCH_= 90

ABC_v

MHC=

HDA_= 90

IHD_(IlgiaoimcaCHvAB).Suyra ADHCHM.DoDHHM=AHMC.VvytacDH=HMAHMC(1)Hontontngt,tacHE=HMAHMB(2)T(1)v(2),kthpviMB= MC,tasuyraHE= HD(iucnchngminh)(ii)Cch2.LyPlmtimbtktrnngthngquaAsongsongviDE(P , A),QlmtimbtktrnngthngquaHsongsongviBC(Q , H).Tac:MBMC= H(BCMQ)vHDHE= A(DEHP).139MHB AC, HC AD, HM AP, HQ AHnnH(CBQM) = A(DEHP).SuyraMBMC=HDHEMMltrungimBCnntaciucnchngminh. Bi2.71 Cho on thng AB= a c nh. im Mdi ng trn AB(Mkhc A, B). TrongcngmtnamtphngblngthngABdnghinhvungAMCDvMBEF.HaingthngAF, BCctnhauN.TmvtrimMsaochoonMNcdilnnht.Li giiNABMD CF E(i)Cch1.GiscchnhvungAMCD, BEFMchngdng.KhiR(M,90):A C, F B.SuyraR(M,90)(AF) = CB.Do

ANB= 90nntgicANCMnitip.Vvy

ANM=

ACM= 45hayNMlphngicca

ANB.Mtkhc,trongmttamgicthngphngicluncdinhhnngtrungtuynxutphttcngmtnh.SuyraMN AB2(ii)Cch2.Tcch1tac

DNM=

ENM= 90.SuyraD, N, Ethnghng.DoMNlngcaocatamgicvungDME.Vvy1MN2=1DM2+1ME2

12_1DM+1EM_2

12_4DM+ME_2=4AB2SuyraMN AB2140ngthcxyrakhiMltrungimAB.VyMNtgitrnhnhtbngAB2khiMltrungimAB. Bi2.72 Cho tam gic ABCnhn khng cn, ni tip (O). Cc ng cao AA0, BB0, CC0ngquyti H. CcimA1, A2thuc(O) saochongtrnngoi tipcctamgicA1B0C0, A2B0C0tipxctrongvi(O)tiA1, A2.B1, B2, C1, C2xcnhtngt.ChngminhrngB1B2, C1C2, A1A2ngquytimtimtrnOH.Li giiXAA0B0C0HOABCA1A2GiXAlgiaoimcaB0C0viBC.TngtchoccimXB, XC.DchngminhcXA, XB, XCthnghngvA1, A2lcctipimcahaitiptuynktXAivi(O).CcimXB, XCcnhnghatngt. Gi lngthngi quaXA, XB, XC. VA1A2lngicccaXAivi(O)nnA1A2, B1B2, C1C2ngquyticccangthngivi(O).chngminhimngquynmtrnOH,tachcnchngminhvunggcviOH.V B, C, B0, C0ng vin nn XAnm trn trc ng phng ca (O) v ng trn Euler catamgicABC.Tngt,suyraltrcngphngca(O)vngtrnEulercatamgicABC.DongthngvunggcviOH.Taciucnchngminh. Bi2.73 Chongtrn(I)nitiptamgicABCtipxcBC, CA, ABtiA1, B1, C1.Cc ng thng IA1, IB1, IC1 tng ng ct cc on thng B1C1, C1A1, A1B1 ti A2, B2, C2.ChngminhccngthngAA2, BB2, CC2ngquy.Li gii141VSMB2C2A2A1B1C1IABC(i)Cch1.(Sdngtnhchthngimiuha)TaschngminhrngAA2iquatrungimMcaBC.NuAB= ACthiunylhinnhin.NuAB ,= AC,giSlgiaoimcaB1C1vngthngquaAsongsongBC.V lgiaoimcaASvA1A2.Khi

AV I=90.SuyraV B1IC1nitip,mIB1=IC1nnV A2lphngictrongca

B1V C1VV S V A2nnV Slphngicngoica

B1V C1.Do(B1C1A2S) = 1hayA(BCM

S) = 1viM

lgiaoimcaAA2viBC.SuyraM

ltrungimBChayM M

.DoAA2, BB2, CC2lccngtrungtuyncatamgicABCnnchngngquytitrngtmtamgic.(ii)Cch2.(SdngnhlMenelausvCeva)DoA1A2, B1B2, C1C2ngquynntheonhlMenelaustac:A2B1A2C1

B2C1B2A1

A1C2C2B1= 1MAB1= AC1nn:sin

BAA2sin

CAA2=C1A2B1A2, sin

ACC2sin

BCC2=B1C2B1C2, sin

CBB2sin

ABB2=A1B2C1B2Suyrasin

BAA2sin

CAA2

sin

ACC2sin

BCC2

sin

CBB2sin

ABB2=A2B1A2C1

B2C1B2A1

C2A1C2B1= 1TheonhlCevadngsintaciucnchngminh.(iii)Cch3.Tng t nh bi 2.44, ta suy ra AA2 l trung tuyn ca tam gic ABC. Suy ra AA2, BB2, CC2ngquytitrngtmtamgicABC. Ch. Cch 1 ca bi ton ny c th p dng cho bi 2.44 nh mt cch chng minh khc.142Bi2.74 Cho tam gic ABCcn ti A. Trn tia i ca tia CA ly im E. Giao im caBEvphngicgc

BAClD. ngthngquaDsongsongABctBCF. AFctBEtiM.ChngminhrngMltrungimBE.Li giiMNFDHBCAE(i)Cch1.GiNlgiaoimDFvAC,dcNA = ND.Tac

DNC=

EABv

DCN=

EBAnn DNCEAB.SuyraNCND=ABAEDoFCFB=NCNA=ACAET , p dng nh l Menelaus cho BCEvi cc im A, F, Mta c iu cn chng minh.(ii)Cch2.GiHltrungimcaBC.TacABDFlhnhthangnnMHiquatrungimcaAB.MMHsongsongviCEnnsuyraMltrungimcaBE(ngtrungbnhtrongtamgicBCE). Bi2.75 Cho tgicliABCDsao choABko song songviCDv imXbntrong tgic tha

ADX=

BCX< 90v

DAX=

CBX< 90. Gi Yl giao im ng trung trccaABvCD.Chngminhrng

AY B= 2

ADX.Li gii143NMO'OZO2O1YBADCXB:Chohaingtrn(O1)v(O2)ctnhautiX, Z.LyAlmtimbtknmtrn(O1).DngtiaZBixngtiaZAquaZXviBthuc(O2).GiOltmngoitipABZ.KhitacOO1= OO2.Chngminhb.(OO1, O1O2) (OO1, AZ) + (AZ, ZX) + (ZX, O1O2) (O1O2, ZX) + (ZX, ZB) + (ZB, OO2) (O1O2, OO2) (mod)DotamgicOO1O2cntiOnnOO1= OO2.Bcchngminh. Trlibiton:Gi(O1), (O2)lnltlngtrnngoitip XADv XBC.GiZlgiaoimthhaica(O1), (O2).Gi(O), (O

)lnltlngtrnngoitip ZABv ZCD.GiY

lgiaoimthhaica(O), (O

).Tac:M= ZX (O)(M ,= Z), N= ZX (O

) (N ,= Z)Tac:(ZA, ZX) (DA, DX) (CX, CB) (ZX, ZB) (mod)nnpdngbtrntacOO1= OO2.TungttacngcO

O1= O

O2.SuyraOO

O1O2.MtkhcXZ O1O2, ZY

OO

nnZY

ZX.144Xt (O) c ZY

ZMv Ml im chnh gia cung ABkhng cha Y

, ta suy ra Y

A = Y

B.Tngt:Y

C= Y

DnnY

Y .Vvy

AY B=

AZB= 2

ADX.Taciucnchngminh. Bi2.76 Chotgicli ABCDni tiptrong(O). ADctBCti E, ACctBDtiF.M, NltrungimAB, CD.Chngminhrng:2MNEF=ABCD CDABLi giiVUPNMGEFBACDGisAB< CD, BC< AD.Gi Pl trung im EF. Khi M, N, Pthng hng (p dng nh l Gauss cho t gic tonphnAEBFDC)vMnmtrnonPN.Trcht,taschngminh2PMEF=ABCD(1)TngngviPMAB=PFCDGiUlimixngviFquaN, V ltrungimEU.TacCFDUlhnhbnhhnh.Nn

ADU= 180

CAD = 180

CBD =

EBF.Mtkhc,tacFBDU=FBFC=ABCD=EBEDDo EBFEDU.SuyraPBAB=V DDC145Tngt,tasuyra PABV CD.TtacPMAB=V NCD=PFCDngthc(1)cchngminh.Hontontngt,tachngminhc2PNEF=CDAB(2)T(1)v(2)tasuyraiucnchngminh. Bi2.77 ChotgicABCDnitipcmtngtrn.Chngminhrng:ACBD=DAAB +BCCDABBC +CDDALi gii(i)Cch1.FEABDCKdyDE, CFsongsongviAC, BDtngng.Tac:AE= DC, CE= AD.pdngnhlPtolemytacEABC +ABCE= ACBETngngviDCBC +ABDA = ACBE (1)Tngt,tacABBC +CDDA = AFBD (2)Chiatheovhaingthc(1)v(2),vichrngAF= BE,taciucnchngminh.146(ii)Cch2.Tac:ACABBC +ACCDDA = 4R(SABC +SCDA)= 4R(SABD +SBCD)= DAABBD +BCCDBDTsuyraiucnchngminh. Bi2.78 ChotamgicnhnABCni tip(O; R).Gi R1, R2, R3tngnglbnknhngtrnngoitipcctamgicOBC, OCA, OAB.Chngminhrng:R1 +R2 +R33RLi giiOAB CGiO1ltmngoitipcatamgicOBC.Tac:R1= O1B,

BOO1= ApdngnhlhmssinchotamgicBOO1tac:R1= O1B=OBsin

BOO1sin

BO1O=Rsin Asin 2A=R2 cos ADoR1 +R2 +R3=R2_1cos A+1cos B+1cos C_

R2 9cos A + cos B + cos C3Rnychngminhhontt. Gia nhng b c thng minh ngang nhau v trong nhng iu kin tng t,ai c tinh thn hnh hc th ngi s thng v thu c mt cng lc hon ton mi m.Blaise PascalHnh hc l khoa hc ca l lun chnh xc trn cc s liu khng chnh xc.George PlyaHnh hc l nn tng ca tt c cc bc tranh.Albrecht D urerCm hng lun cn thit trong hnh hc, cng ging nh trong thi ca.Alexander Pushkin