DIEM c0 D!NH.
*fei tofn 1. Cho dadng trdn (O; R) vd m6t
d.rctng thang a cat p) tqi C, D. M,t di€m M di
dQng ftAn fia d6i cila tia DC. Qua M kd haiti€p ruy€n MA, MB vdi dúdng rrdn (O) (A, B ld ti€p ruy€n MA, MB vdi dúdng rrdn (O) (A, B ld ti€p di€m). Chtmg minh dadng thdng AB lu6n
^ =.; : _
di qua mor diem co dinh.
Ldi girtL Gqi Hld trung di€m CD virgiao di€m
cila AB vu MO, OH ldn luqt ld E, F (h.1).
F
I
Tam gi6c OBM vu6ng tpi B, dulng cao BẸ
Suy ra OẸ OM: OB2 : R2 (1)
Do Ffrfu =FEfu =900 n6n tb gi6c MEHF
nQi ti6p. Tti hai tam gi6c vu6ng OHMvd OEF
đng dpng, suy ra
oH
=oM =oF =oÉoM (2\
OE OF OH
Tt (1) ve (2) suy ru OF =
Do dulng trdn (O), ctuong thhng d cho tru6c, n6n Ollkh6ng tl6ị
Suy ra OF kh6ng dOi, di6m ,f'cO einfr. Do d6
tluong thhngAB di qua di6m F cO Ainfr.O NhAn xdt: Bilito6n vdn dring trong trucrng hqp
di€m M nim f6n tia tlOi cua tra CD. Khi d6
dunng thfurgAB v6n di qua iliiSmf cO einn.
*gei lnhn 2. Cho fudng trdn tdm O vd hai di€m A, B cA dinh thuQc dtrdng trdn do (AB kh6ng phdi ld &.rong kinh). Gei M la di€m
chinh gitta cr)a ctmg nhd TẸTrAn dọan AB Itiy hai diim C, D phdn biQr va khong ndm tr€n
fudng trdn. Cdc drdng thdng MC, MD ctitdtrdng tron đ cho tr.rong ịmg tqi E, F khdc M. dtrdng tron đ cho tr.rong ịmg tqi E, F khdc M.
l) Ch*yS minh rdng b6n di€m C, D, E, F
citng ndm ftAn mot dtrdng trdn.
2) Gqi Ot, Ọ: tLrong irng ld tdm cdc dtrdng trdn ngoqi ti€p tam giac ACE va BDF. Chimg minh rdng.khi C, D thay d6i tAn dọqn AB cac dadng .thiing AO1 vd BOz luon cdt nhau tai
m6t di€m co dinh.
LN gi,rtL D Xdt fiudng hW C nir* gi*o Avd D (h.2) Hinh 2 R2 OH t_ C6 MCB:a1sd MB +sdAE), 2' I^
MFE ==1sdU,l +sdAE)
2',
r.ait sdfu : sdff4, n€nfuE = ffiE .
TOfiN HQC eee*6ae&s6se686**&es&*as&6a*s{
C6 frE +EeE = 1800, suy ra
6dE+ ffiE=1800. Suy ratft giitc CDFỆ..t ^...t
nor uep.
Xdt trudng hW D ndm giiia A vd C.
Ta ctng chimg minh duoc C, D, F, E ctng
ndm tr€n mQt dudng trdn.
2) Hq OtHIAC, c6 OA: O€ > A,OAC
^
cdn tpi Or > AqC :2. AqH .
rn trR : 2.FEa - @E : Gd . rvti,
fna : fua ncn frrr : fuE .
Xdt LAOflyudngtqiHcoTorH + ttlor:9go TorH + ttlor:9go
* fun + ilo-r:9oo = ffijr:900, do
db MA ld tiiip tuyrin cria (O1). K6o đi AOt cit
(O) tqi N thl M, O, N thinghdng vi Nh di6m
chinh gifra cung 16n 7E . Up lufn tuong t.rr
BP di qua N ld di6m chinh gita cung l6n
AB . Do d,6 AOr, BO2 di qua di6m N h di6m chinh gifra cung l6n D.
Trudng hqp D ndm gifra A vd C tln AOt vir
BO2 cfrrng fi qua N. Vay AOy BO21u6n di qua
mQt di6m c6 dinh (1V). tr
*nai to6n 3. ;^ Cho tam gidc ABC vd diint D di chuyen rren canh BC (D khac B va Q
Dudng trdn (O) di qua D vd ti€p xtic AB tai B.
Ducntg trdn (O2).di qua D vd ti€p xuc AC tai C.
Goi E ld giao di€m tha hai cua (Ot) vd (O). a) Chlmg minh rdng khi D di dong rren doan BC thi dactng thdng ED lu6n di qua m|t di€m
c6 dinh.
$ KA qua trOn cdn dung kh6ng trong trrdng
hW D di dQng 6 ngodi doqn BC.
Ldi gidL a) Ggi (O) lil duong trdn ngoai ti6ptamgi6c ABC (h.3). tamgi6c ABC (h.3).
Hinh 3
c6 FEa =fED;frB =GD . Suy ra
^ ^ -^
BAC + BED + CED = BAC + ABC + ACB = 1800
Do 116 t0 gi6c ABEC nQi tirSp.
Gi6 sri DE chttludng trdn (O) tai <1i6m thri hai
^
S. Tt ABC = BED; n€n hai cung AC vir SB
bing nhau, do d6 S h di6m cO einn.
b) Truong hqp diem D nim ngo di doan BC. Chdng hqn D ndm tr6n tia d6i tia CB Qr.4). (truong hqp D thuQc tia tlOi tia BC chimg minh tuong t.u).
ẠS
Ot Hinh 4
Ta chimg minh tluoc b6n di6m A, B, C, E
cirng nim tr6n dulng tron (O). Gii sri D,E cfit
(O) tqi tIi€m thri hai S, k6 tia Cy ldtia d6i cria tia CẠ Khi d6 trong dulng trdn (O) ta c6
CED = DCy; DCy = ACB .
Suy ra GD =frB q*ong doi;.
^
Tt d6 SEC = 1800 - CED (l$6ng t16i). NCng6c SEC kh6ng OOị V4y di6m S cO Ainfr. O g6c SEC kh6ng OOị V4y di6m S cO Ainfr. O
*nei lr6n 4. Cho goc vu6ng xAy, di€m B c6 dinh tAn Ay, di€m C di chuy€n tuAn Ax.
Dactng trdn tdm I nQi ti€p tam giac ABC ti€p
xtic vdi AC, BC theo tha ttr o M, N. Chang
l-
minh rang drdng rhdng MN lu6n di qua m6t di€m ctj dinh.
Ldi sidị Gqi H: AI n MN (h.s).
v
B
,Ạ Hinh 5
TOAN HOC &asse66€qesa&aa6e€&s&&es$sea*€
Yl CM: CN, n6n tam gi6c CMN cdntai L.
t^
Suy ra eNu =9oo -:C.
2
Do d6 6iE =e00 +1a
2
Do 1ld giao di6m c6c tludng phAn gi6c trong l^
ctra tam gi6c ABC,n€n BIA = 900 +:C
2
Do d6 BIA= BNH . Suy ra th gi6c BIHN nQi
ti€p. Lai co BNI = 900 + BHI = 900. Do d6
tam giitc ABH vu6ng tai H, lai c66ZE =450, suy ra tam gi6c ABH r,udng cdn 6ZE =450, suy ra tam gi6c ABH r,udng cdn tqi H.
Do A,.A ci5 dinh, n6n dii5m rr co oinh. vay
trDllu6n di qua di€mIlcO dinh. B,^ ,^
NhQn xit. Trudnghqp t6ng quit xAy = a th\
tam gi6c ABH vu6ng tqi H, AB cho trudc,
87il =a . Sry ra tti€m A cO einn.
2
*nei todn 5. Cho drdng trdn tdm O, đ1,
cung AB. DiAm M di chuyAn ftAn cung ktn
AB. Cdc dudng cao AE, BF cita tam gidc
ABM cdt nhou d H. Dadng trdn tdm H bdnk{nh HM ciit MA, MB theo thth tr d C, D. k{nh HM ciit MA, MB theo thth tr d C, D.
a) Chtmg minh riing durt s thdng ke tii Mvu6ng
g6c voi tO fuAn ilịqua m\t di€m.cti dinh. A1 Ct
""g minh ring doons thdng ke tir H vd
vu6ng gic voi CD cilng di qua mQt diAm cd dinh. Lirt girtị a) K6 tia ti6p tuytin tuIx vbi clucrng
-,..^
trdn (O) (h.6). Ta c6 My: MAB .
Hinh 6
Tri gi6c ABEF n}i tiiip tluong tron dulng
kinhAB,n€n MEF = MAB .
Do d6 MEF = Mr, n€n MxllEF. Suy ra
OMLEF. Ta c6 H ld tdm duong trdn ngo4i ti6p tam gi6c MCD, HE L MD,
"9+ n
P !ru"g
di€m MD. Tucrng tU F h trung ili6m MC n€n
EF lil tlucrng trung binh tam grhc MCD. Dod6 EFllCD, md OMLEF. Suy ru OMLCD. d6 EFllCD, md OMLEF. Suy ru OMLCD.
Vfly di6m cO Ainft n,Ọ
b) Gqi K ld di6m d6i ximg v6r O qua AB, ta c6 OK L AB, md MH I AB. Suy ra MHll OK.
Lai c6 trong tam gi6c khoing cbch tu tr.uc
t0m tam giic d}n <tinh bing 2,16,n Jchonng
c6ch tu t0m dudng trdn ngopi ti6p d6n cpnh
tucrng img. Do d6 MH : OK, suy ra tb gi6c
MHKO 1d hinh binh henh = HKllOM, mirOMLCD, n€n HKLCD. YQ,Ỵducrng thing OMLCD, n€n HKLCD. YQ,Ỵducrng thing
ke ft H vu6ng g6c CD tli qua di6m K.
Do O, AB cho trudc, n6n K ld tti6m c6 dinh.tr
NhQn xit: . Trong phdn a) dlra vdo tfi giitc
ABEF nQi ti6p dulng trdn, dg do6n duong
thing đ cho tli qua di6m O cO dinh. .
. Trong ph6n b) d6 dUa vdo tinh chdt trong
tam gi6c khodng c6ch tu trgc tAm tam gi6c
d6n dinh bdng 2 lAn khoing cSch tu tdm
ducrng trdn ngopi tiCp diin cpnh tuong img.
*gai toin 6. Cho tam gidc ABC, M ld di6m
biit ki thuQc dudng trdn (O) ngoqi ti€p tam
giac iiỵ Gp.i D ld diAm diii xtug voi M quaAB, E ld diAm đi xilmg vrti M qua BC. Chilmg AB, E ld diAm đi xilmg vrti M qua BC. Chilmg
minh rdng khi di€m M di chuy€n tAn dtdng
trdn (O) thi DE lu6n di qua mot di€m cd dinh.
Ldi gidl Gqi H, 1, K theo thu t.tr lir chdn c6c
duong vu6ng g6c k6 ft M d€'r AB, AC, BC.
Ta c6 H, I, K thing hdng (bpn dqc t.u chimg
minh). Ggi N ld trUc tdm ctra tam gi6c ABC
(h.7). D
qh7 E
Ldi gidi Gi6 str ducrng trdn ngoai ti6p tam
gi6c MPF vd MQE cit nhau t4i M,N. Duong
th[ng MN cdt PQ, BC theo thri t.u tai K vd 1
AN cilt Q) tqi F. Ta cb Edfi =6G, suy ra
BC IiL trung truc NF, md BC lil trung tryc cria
MẸ Suy ra i=A=fr. Tri A--e,
-,^
Kt = Cr, suyra Kt = E, do d6 NEllHK.
Chtmg minh tuong W c6 NDllHK.
YAy,D,N, E thinghing, ndn DE di qua diiSm
N cd dinh. tr
NlAn xit. + Ducrng thing di qua H, I, K
thlng hing ggi ld đng thdng Simson.
+ Duong thdng di qua 3 di6m D, N, E ggi li
fudng thdng Steiner.
*nai b{n 7. Cho d,rong trdn tdm (q. f*
di€m A ,ij dinh 0 ngoiti (O) ke ti€p tuy€n AB,
AC toi (O) (8, C ti€p diA@. Lay di€m M trAn
cung nhd BC. Gpi D, E, F thtr ta lạhinh
chi€u tịc M d€n BC, AC, AB. Goi MB cdt DF
tqịP, M ,C ciit DE tq.i Q. Chtrng minh dwdng
thdng n6i giao diAm cia hai &rdng trdnngoqi tidp tam giac MPF va MQE lu6n di Eta ngoqi tidp tam giac MPF va MQE lu6n di Eta
m)t di€m co dinh.
Tucrng W KP ld ti6p tuy6n ctra duong trdn
ngoai ti6p LMPF. Ta c6 KM. KN : KA,KM. KN:fP. Suy ra KP: Ks.Xdttamgiitc KM. KN:fP. Suy ra KP: Ks.Xdttamgiitc
MBC, PQllBC, KP.: X8. Theo dlnh li Thales,
sly ralld trung di6m,BC. YQy MN di qua di6m
c6 dinhlld trung diOm criaBC. O .