qua BC ctra giao dii5m
DF vit AB chinhldF,
vdy F,M tl6i ximg qua M M
BC, K
Tucrng W E,N e6l xtmg qrura BC. VQy S ldgiao cira BN vd CM sE ld di6m tltii ximg qua giao cira BN vd CM sE ld di6m tltii ximg qua
BC cira T ld giao cira BE,CF . Ta c6
DET = DAB = DCF n6n tu gi6c DTEC nOi
ti6p. Tri d6 TDC =TEA= BDA, suy ra DT vit
DA d6iximgqua BC, ndn DT diqta K hay
D,K,T th[ng hdng. Suy ra 6nh dOi ximg qua
tryc BC clioa D,K,T ld D,A,S thing hdng.
Bii 7. Cho tam giac ABC nQi fiAp du'rlng tron
(O). Phdn gi,'rc Eei cat @) tqi D khac A.
Liiy X thu6c AC sao cho OXll AD. Y td dii:n
dtit xung voi X qua A Lifu Z tr€n tia AB sao cho AZ = AC Ldy T thu6c AO sao cho Zf )- AC .
J-
ChLrng minh riing du'rlng trdn du'dng kinh YZ
L'htu Llot doun l)1.
(TiAp theo k) tru'6c) l TRAN QUANG HUNG (GYTHPT chuyAn KHTN,DHQG Hd Ni.i)
Ldi gi,rti. Gi6 str OX cat AB tqi n . O€ thay
LAXE citn tqi A n€n AY = AX = AE , suy ra I
vd E dOiximgnhau qraAD.
E
O6 tnay Z vit C dOi xtmg nhau qua AD, ggiI4z ld di6m dOi xtmg voi B qula AD thi tam I4z ld di6m dOi xtmg voi B qula AD thi tam
gi6c ABC vd AWZ dOi xtmg nhau qua tryc
AD . Do ZW vir BC ddi ximg qua ph6n gi6c
BAC n€n AOLZW, trr d6 7 ld tryc tAm
LAZW, v1y T dOi xtmg v6i tryc tdm H oiaLABC qua AD. Ggi S,F lAn luqt li trung LABC qua AD. Ggi S,F lAn luqt li trung cli6m cira DT,DH thi S,F d6i xtmg nhau qua
AD. Ta cAn chimg minh fr7= 900. Qua phdpAOi ximg trvc AD cAn chimg minh iFi = 900, AOi ximg trvc AD cAn chimg minh iFi = 900,
that vay goi DG ld cluong kinh cira (O). DCthiy OEIIAD IAG n6n EA=EG. AG ldph6n thiy OEIIAD IAG n6n EA=EG. AG ldph6n
gi6c EAC n€n EGA= EAG =GAC = EGllAC.
Gqi CE cht (o) tai P k}lhc C vd CE cbt lGt1i Q . Ta th6y PGQ= PCA= PEG stry ra t1i Q . Ta th6y PGQ= PCA= PEG stry ra
te nr,,r-ror., T?[ilrHEE 31
MGQaLQEG =Y=q=*.- GE QE CE.
_- .. PG GE EA AP PG AG
ludo
AG= cE= cE= BC = PA= BC (r)
Ggi L,M lAn luqt li trung di€m AD,BC d6 c6.I .I
OM vir FL ctng song song vi bdng
)AH nenFMOL h hinh binh henh. Suy ra FMOL h hinh binh henh. Suy ra
FM =OL=!AC. Z , PG AG zFM FM rir(l) "6 A=#=ffi=ffi (2) DE c6 trFd =$oo -6G =Ffri (3)
Tt (2), (3) suy ra MAG@ LMCF suy ra
-^ ^
PGA= MFC . Tn d6 gqi CK ld clucrng cao vdN ld trung di6m AB ta thhy ttr gi6c KNMF N ld trung di6m AB ta thhy ttr gi6c KNMF
nQi ti6p dudng trdn Euler c;0ra LABC, suy ra
ECF = ECA+ ACB+ BCF
^ ^
= PGA+ NMB + BCF = NMB + MFC + BCF
=fifrA*fifrF =ffiF = 18oo -frEF.
Tird6tu gi6c EKFC nQitirip, ,uyruiFe=900.
NhQn xdt. ViQc dimg diii xilmg truc AD dd chuy€n m6
hinh bdi toan qua viQc chrimg minh 6Fi =g0o ld m\t vi€c ldm qudn treng vd vai trd cila viQc su &1ng phip
adi xnng tnlc 6 day ld khd n6t bQt. Ta dilng di5i nimg truc chuydn todn b0 m6 hinh bdi todn nay thdnh mQt bdi todn khdc ld mQt cdch gidi dfic trwng cho viQc su
&1ng phep bien hinh trong gidi toan.
Bdi 8. Cho tam giac ABC
"ii dinh voi tdm
dudng trdn ngoai ti€p O vd phdn giac trong
AD. P, Q ld hai di€m ddng giac trong tam giac
ABC vd niim tAn AD. Liiy didm S sao cho
QSIIAO vd DS LAO. Gqi M ld trung di€m
cila BC. Chrmg minh rdng dudng thdng qua P
vu6ng goc vdi SM ludn di qua m\t di6m
"6 dirh
khi P, Q thay diii.
Ldi gi,fiL Gqi R h hinh chii5u cira Q trdn BC.Do DS IAO n€n DS vd BC ddi ximg nhau Do DS IAO n€n DS vd BC ddi ximg nhau
qta AD, tt tl6 aE m6y R vi S d6i ximg nhauqua AD. Ggi N,I ld c6c di6m d5i ximg v6i qua AD. Ggi N,I ld c6c di6m d5i ximg v6i
M,O qlira AD tuong img. Ta sE chimg minh
PT LMS, hay <ludng thing qua P vudng g6c v6i SM tli qua r c5 Ainl. X6t qua phdp <I5i ximg tryc AD th\cin chimg minh RN LOP.
F
E
ThQtvfly gai AD cit (O) tai E thlc A. EF
ld duong kinh ctra (O). M ld trung dii5m ctraBC. H ld trung <litim cira MN thl 11 thuQc BC. H ld trung <litim cira MN thl 11 thuQc
^
AD . Ta c6 RMN = OEP (1). Theo dinh 1)?
Thales vir chri f N{DM cn LMDE ta c6
2! RM DM HM MN MR = 2+ = Y':= ='I-!, = YI ='!=' QE e)
^
Lai c6 PBE = PBC +CBE - QBA+QAB - BQE .
Tri d6 c6 A,EBP a LEQB + EP.EQ= EBz =
= EM.EF =zEM.Eo *zY=E 8E =ZEO (3)
rt(2), (3) = ,*W=# @)
Tr (1), (4) * MPOo LI1INR -ffin=6Fd.
Gii sir RN cit OP t4t K thi tu girc PHNK nQiti5p, suyra frEp =fifi =900. v6y RN LoP. ti5p, suyra frEp =fifi =900. v6y RN LoP.
NhQn xit. Yi€c sth &tng phdp diii xung tryc AD chuy€n bdi todn vi ch*ng minh RN LOP h mAt
viQc ldm quan trong trong bdi todn ndy.
Ta tii5p tpc vcri bdi thi HSG lorp 10 o TrudngTHPT chuy6n KHTN, DHQG Hd NQi ndm 2015 THPT chuy6n KHTN, DHQG Hd NQi ndm 2015
Bni 9. Cho LABC nhqn, cd dinh. P ld didm di
chuy€n tuAn csnh BC. (K),(L) ldn laqt ld
dwdng trdn ngoqi ti€p cdc tam giac PAB,PAC.
Liiy S thuQc (K) sao cho PSIIAB, tay T thuQc
lL) sao cho PT llAC
a) Chimg minh riing dadng trdn ngogi ti€p tam
giac LAST lu6n di qua m1t di€m J c6 dinhkhac A. khac A.
,.^ TORN HQC
b) Gia sw (K) (:fu CA tui E khac A ({-) cat AB rei F khac A. BE c& CF qi G. Chu'ng minh rdng
furi;ng thcing PG di qtta J <> AP di qua tdntdu'rmg trdn Euler ct)a taru gidc A,ABC . du'rmg trdn Euler ct)a taru gidc A,ABC .
H
LN gidi. a) Gqi ,I ld tim <lucrng trdn ngopi ti6p
LABC. Ta sE chimg minh <ludng trdn ngo4ititip AASil di qua f cO ainn. Th4t vfy, gqi titip AASil di qua f cO ainn. Th4t vfy, gqi
AT,AS giao BC tqi Q,R tuong img. C6c tfi
gi6c ABPS vd ACPT ld hinh thang cin, do d6
tam gi6c ABR vir ACQ cdn tai R vd Q.DE
thdy JK,JL 16n luqt ld trung tr.uc cria AB,AC, do d6 JK di qua R vit JL tli qua Q. Suy ra
trong MQR thi ,f h tdm duong trdn nQi titip
hay AJ ld phdn gi6c fu (1). Tf gi6c ACPT
ld hinh thang cdn ndn lQ cflng li trung tr.uccl;;a PT, do 116 JT = lP . Tuong tv JS = JP . cl;;a PT, do 116 JT = lP . Tuong tv JS = JP .
Do d6 lT = lS (2) . Tt (1), (2) suy ra .r thuQc
clucrng trdn ngopi tiiSp AAS".
b) Ta thdy tfi gi6c AEGF ndi ti6p= fri =
=EtrF =fFd n6n tu gi6c CEGP nQi ti6p k6o
theo tu gi6c BFGP cflng nQi ti6p. Ggi 11 ld
di6mdOiximgv6i A qua BC thlilFfr=frD=
=AEB =GPC, do d6 PG di qua H.Do d6 PG
vd AP dOi xtmg nhau qua BC. Tt d6 PG di
qua tdm duong trdn ngoai ti€p J cin MBC
e AP tli qua di6m dOi ximg ctra J qua
BC e AP di qua tdm iludng trdn Euler criaAABC. AABC.
Bei 10. (NI& rQng bhi toin 1 thi IMO 2016).
Cho LABC cd ABC til vd tdnt d$ng trdn ngoai tidp ld D. Drting tnmg tt'qrc c:ua AB cdt AC tqt l.
K,M ldn htot ld tdm du:dng trdn ngoqi ti6p cac tcun giac ADF,BFC . Dtptg hinh binh hdnh AL[Yl'i. Chung rnink ring Rd/l, FX, BD ding qtr.1'.
Ldi gidi.Do DA=DB vit FA=FB ndn A,B
doi xtmg nhau qua FD . Tt d6 fED -f,tri:
= DCF n€n tu gi6c FBCD nQi ti6p dudng trdn
(M). Hai clunng trdn (BFDC) vit (AFD) c6MFD = LBFD (c.g.c) n6n b5n kinh cta chring MFD = LBFD (c.g.c) n6n b5n kinh cta chring bing nhau. Tir d6 MFKD ld hinh thoi, suy ra
K,M d6i ximg nhau qua FD. Y6,y ABMK lir
hinh thang cdn, md AMXK li hinh binh hdnh
n€nd6suyra B,X dliximgnhau qua KM.Tir
MFKD ld hinh thoi suy ra F,D <l6i ximg nhau
qlcra KM, dod6 FX,KM,DB tl6ngquy.
Bni 11. Cho LABC v6'i tdnr clu'ong trdn n6i ti6p I.
P tlnfit: canh AB sao cho IF .L IB . R,S tdn tu'o1
lii trung di€m IP,E.Q la hinh chi1u cr)a I tAn
FC. Chtng ntinh rdng.4Q c:hia d6i dogn RS.
Loi gi,rti. Ggi K,L l6n luqt Ld cic cli6m tl6iximg vdi 1 qua A,Q. Ta sE chring minh KL ximg vdi 1 qua A,Q. Ta sE chring minh KL
chia d6i PC. ThQnvQy.
[-B AEi
ptC = ptA+ AtC =
]l+90" * ";-
=180" -Bfc =FIk. Lai c6 AP =AP -PI .
2 . -" AK AI IC,dlng thfc cu6i c6 do LAPI.n LNC. Tt d6 dlng thfc cu6i c6 do LAPI.n LNC. Tt d6
MPKurNrc. Liy H tr€n PC sao choAHllBC. .,^ PB PB PA PC PI PI Ia ilav ma PK_ PA.PK PH.PC_ PH" KPB = DPH n€n MIH a MBK suy ra LPHK u) LPIB, n),n ffrF = 90". Ggi T le -..1 -i.
cli6m d6i xung vdi K qua H thi