gi6i bdi to5n ndy. Ldi giii cira hai b4n c6 y nhung
chua dugc hodn chinh.
DANG HIING rHANc BitiTl2l457. Cho tam giac ABC nQi ti€p dadng trdn (O) vd ngoqi ti€p daong trdn (I). Cac ti€p
nydn voi (O) tqi B vd C
"iit ,hou tqi T. Goi Mld trung di€m cila BC; D ld di1m chinh gitra ld trung di€m cila BC; D ld di1m chinh gitra
cung BC kh6ng chira didm A; AM cdt lqi (O) tqi E; AT ciit cqnh BC tqi F; J ld trung didm cila
dosn IF. Chilmg minh rdng m = m.
Ldi gidi. nA ai. Cho tam gidc ABC nQi ti€p
dudng trdn (O). Cdc fiAp uyiin vdi dudng trdn (O) tqi B vd C cdt nhou tqi T. Chilrng minh riing
AT chilra dwdng diii trung crta tum gidc ABC kd
t* dinh A.
Chilmg minh. (h.1) Gqi MD trung di6m cilr- BC; N ld giao <Ii6m thri hai cria AT vor tlucrng trdn (O). Vi ABNC ld nir giac diiu hda n€n
BN.AC: AB.NC (1)
Su dung dinh lj, Ptolemy cho tu giSc nQi tii5p
ABNC tathdy
BN.AC + AB.NC : AN.BC (2)
A
T
Hinh I I
Tn (l) vd (2) suy raBN.AC : ;AN.BC : AN.CM
I
- BN AC .:.
hay ffi=ffi , k6t hqp voi ANB = ACM ta
thiy MBN ctt MMC (c.s.c)
= 6trfi = ffia .
Tri <16 luu V
"*rg Mldtnmgdi6m ctra BC ta suy ra
AN, hay AT ch?a du'ong ddt mmg cria tam gi5c
ABCketu dinhl. nO CIA duqc chimg minh.
Trd lqi bdi todn (h.2).
A
T
Hinh 2
Tri gi6 thi6t bei ratathdy AFld duong diii trung cta tam gi6c ABC (Theo gO 0d). Do d6 AF vit AMlithai cludng <15ng gi6c d6i vdi goc 6Aa .
Gqi I, liL tem <lucrng trdn bdng ti6p fuong irng
v6i <linh A c:ira tam gi6c ABC th6 thi ta c6 k6t O
"(;)' +1<y(n) =y(n+t)=(n+t)'\,)
qua quen thu6c sau : DI : DB : DC : DI,. Suy ra DJ ld dudng trung binh cria tam gi6c
IF 1,. I(]:ri d6 ADJ = AI,F. Ta sE chimg minh ,i"g trF :18i. rhflt v6y, do FEF = frd;
BAF =EAC, n€n LABFct>LAEC (g.g), suy ra
AF AB
id =fr hay AE.AF: AB.AC (3)M{tkh6c fu.=fu +@ =ry "+=fu. M{tkh6c fu.=fu +@ =ry "+=fu.