Mai Lud.n, 10T, THPT Hing Vtrong; Vinh Phric:
Trd.n Vd.n Huynh, 10A1, THPT Ng6 Gia Tq, Lap
Th4ch; HA N6i: Dod.n Tri Dfrng, 10A2, PTCTT-
DHSP I ; Hdng Ydnz Phan TiAh Dang,10T, chuyOn ;
H6i Dtfdng: Vfi Dinh Qu!,10A, THPT Nam S6ch;
Thanh H6at Hod.ng Dtc f, 98, THCS Trdn Mai
Ninh, Phorn Khd.c Thanh,1081, THFT TriQu Son I; NghO An: DSng COng Vinh, 701, THPT Nghia Ddn;
TP Hd Chi Minh: Phan Dttc Thdnh,10T, THPT Le
H6ng Phong; Phri YOn: NguyAn Tudn Drtng,10A, PT
c{p z-e Phan Chu- Trinh, XuAn- L6.c, So]s Cdu;
Kh6nh Hdaz NguyAn Thi Hodng Oonh, 10T, THFT
LO Quf D6n, Nha Trang.
NGLTYENMINH DUCBei T11/337. MOt &tong thiin,g di qua tdm. Bei T11/337. MOt &tong thiin,g di qua tdm. dtons trbn ndi tidp tam giac ABC, cd.t cd.c
canh AB ua AC theo tht tx tai M ua N, Chttng
minh ritng; ,l
Nhi6:u ban chi chrlng minh mQnh d€'mOt chi6:u :MN
di qua / = (*), mh kf,Ong nhAn thdy r6ng, cdn phAi
ch(tng minh mQnh d6'hai chi€:u : MN di qua.I <+ (*).
M c6 chi6:u ngrrQc 14i : (*) =+ MN di qua 1, ta mdi c6thdn6irdng:"KhiAM ='!9 vi.4N= 249 - 2AB - thdn6irdng:"KhiAM ='!9 vi.4N= 249 - 2AB -
BC BC BC,
(*) dring. Suy ra : MN di qua I'. Chi tiet nDry riit quan irong tiong t<hi giai quy61 vdn dd "Ddng thirc xiy ra khi nAo". Ay vAy md nhi6:u ngrrdi r{t hay qu6n n6.
2) C6c ban sau dAy c6 ldi g:Ai tddng ddi tdt:
B6c Giang: Pham Vd.n Tnring. 12A10,, THPI Ng6
Si Li6n, TX B6c Giang; Hi TAy: Nguydn Thd'Tdm,
8A1, THCS Ns6 Si Li6n, Chrrong M!; Nam Dinh:
NguyAn Qudb Eai,9A7, THCS Trdn Ding Ninh, TP
Nim linn; Thanh H6at Dtong Anh. Tuan. THPT
Hau LOc II, HQu l.Qc; Vinh Plrl(ucz Trdn Vdn Huynh,
10A1, THPT Ng6 Gia ft, LAp Thach, ViVdn Quang,
tZAl, Ii COng Truydn,-11A1, THPI chuyOn Vinh
Phric; Phri Thgz Nguy1n Tiah Thanh, 9C, THC,S
Nguy6n Quang Bich, Tam N6ngr Hir NQi: Nguln
Trgng NhQt Quang,11 To6n, DHKHTN.
NGTIYEN }II}JH HA
Bai T12l337. Cho hinh tit diin ABCD. Goi
A j, B t, C 1, D 1 ttin lxm ld, trong tant ccic n'Lot d6i
dlQn uoi cac dinh A, B, C, D. Cac drtottg thfury AA1, BBt, CCl, DDt cdt mdt cdu ngoai tiep ilr
diAn ABCD ldn nlta theo thtl tx tcti A^ B, C,
ir. Ct ,tng minh rdng.'
AAt
+BB1 , CCt _DDt .9 ,*,
AA., BB. C'C',, DD.3
Ldi giai. @r-ta theo Nguydn Vdn Hdu,72Al,
THPI YOn Dring 2, B{c Giang).
Od tnnn bdy ldi giai duqc gon ging, ta thay
ddi ki hi6u c6c didm A, B, C, D, A1,81, C1, Dr
vd, A2, 82, C2, D, theo tht hl ld A1, A2, As, A+i
M1, M2, Ms, M+ ve Ai , 4,, A;, A; . Xhi d6
BDT (*) dttgc vidt gon I4i du6i d4ng :
,=$ 4*''9.
' 7=t A,A', 3
("*)
Goi G lir trong tAm ti1 di6n ArArAsAa , O vit
R lAn ludt lA tAm vd b6n kfnh mit cAu ngoai
tidp tit diQn A,,tArAn ;thdthi ta c6:
BM,CN . BC,AM,AN - 4.AB.AC AM,AN - 4.AB.AC
Ldi giAi. Goi 1
lh t6m, r Ih b6n kfnh drtdng trdn
nQi tidp MBC.
Ta thdy t tj
MN di qua / (hinh vO)
€ Sruu+Sre-nu =ffi.tou"
o lrnv*lr.AIrJ =AI\4'AN .!r( AB+AC+BC)
2- 2 AB,AC2 AB.AC AB,AC cr -+-- AB+AC+BC AN AM AB.CN AC.BM e' --- =B( AN AM AB.CN AC.BM BC,AN BC,AM Theo BDT Cauchy MacBM-cN 1 > 2. 1---__--:- . suv ra \ NC, ,AM.AN BM.CN BC2 AM,AN _ 4AB,AC
D&ng thrlc xAy ra khi vh chi khi
AB.CN AC.BM 1-=- -=- =- BC,AN BC,AM 2, AN 2AB CN BC AM zAC BM BC GA\ =d vd /,@,h= -GAi.GA':= 4 I \
Nh4n x6t. 1) C6 nhiOru b4n tham. gia giAi bdi ndy'
Tuy nhi6n, kh6ng b4n nAo giAi quydt hoen chinh vdn
ad ioing thrlc xi! ra khi nd6 ?". Xin phAn tich cg thd :
24
24
bhrlc sau (sr} dung vectd) :
f*o? =ie$ = 4( Rz -ocz ) = 4p ; (z)
i=1 i=1.