toán nâng cao và các chuyên đề hình học 9

P h i i n th* "hiit .

Chudng I

HE T H ~ C
LllgNG TRONG TAM GIAC V ~ O N G
61.

MOT ~6HE THUC LUQNG VE CANH VA D U ~ N CAO
G TRONG
TAM GIAC V U ~ N G
---

-

Trong mot tam giic vuBng, binh
phuang mdi canh g6c vuang bing tich c i ~ a
canh huy&n v6i hinh chi& cua canh g6c
vuBng do tr&ncanh huy&n.
R -_ - - _
H _ _ - - - - - _ _c
b2 = ab'; c 2 = act

I

Trong m&t tam giic vudng, binh
phuung d u h g cao irng v6i canh huy&n
bhng tich hai hinh chi& cha hai canh g6c
vuBng tren canh huy&n.
h 2 = b'. c'

A
--- ---_------B

Hit~lzI

I

~ r o n gmot tam giic vuBng, tich hai canh g6c vuBng bing tich c i a
canh huy&n v6i duirng cao tuung irng.
Trong m&t tam gi6c vuBng, nghjch dao binh phuung d u h g cao img
v6i canh huy&n bing tBng c i c nghjch d i o binh phuung hai canh g6c
vu8ng.
I - 1 ' 1
-=
+7
h2 b2 c-

V i d!r I . Bigt ti s6 hai canh g6c vuBng c i ~ am&t tam g i k vu8ng lh 5 : 6,
canh huy&n 18 122cm. Tinh d& dhi hinh chigu cha c6c canh g6c vu8ngYren
canh huy8n.


.

_

..

Giii

.

Gih sir tam giiic ABC vu6n&%
, c6
AB: A C = 5 : 6 v i B C = 122cm(h2).
A
AB AC
ViAB:'AC=5:6n&n-=-=k
5
6
suy ra AB = 5k, AC = 6k.
Tam giAc ABC vubng 13 A, theo djnh I$
. .
Py-ta-go, ta c6 :
Hinl, 2
A B+
~A C =
~BC~
hay
(5k)2 + (6k)' = 1 2 2 ~suy
, ra 61k2 = 122', do d6 k2 = 244, suy ra k = 15,62
Vay AB = 15.62.5 = 78.1 (cm)
AC = 15,62.6 = 93,72 (cm)
KA AH IBC. Theo he th* lu(mg v6 canh g6c vubng v6i hinh chi&ucha n6
tr&ncanh huy&n,ta c 6 :

.

A B ~a-=
78,12

AB2 = BH. BC, suy ra BH = BC
122

6099,61
122

w

50 (cm)

A C ~=-=
93,722 8783.44 z 72 (cm)
AC2 = HC. BC, suy ra HC = BC
122
122
Td iiri :DQ dhi hinh chi& c i a ciic canh g6c vubng tren canh huy&n1s i
BH=50cm;HC=72cm
Vi & 2. MQt tam giiic vubng c6 canh huy&n
I i 6,15cm, d u h g cao h g v6i canh huy&n l i
3cm. Tinh cdc canh g6c vubng cha tam gihc.
Gitii
Gih sir tam gisc ABC vubng 6 A, c6
BC = 6.15 cm v i d u h g cao AH = 3 cm (h3). B
11
6-15 - - * Theo he thirc l u w g v& d u h g cao v i hinh chi&u
cha c i c canh g6c vubng tr&ncanh huy&n,ta c6 :
Hirrh 3
AH^ = BH. HC

,/n

-.- --

hay 32 = BH(6,15 - BH), suy ra B H ~ 6,15BH + 9 = 0

o (BH - 3,75) (BH - 2,4) = 0

-- c


G i i su AB < AC, thg thi BH = 2,4 cm, khi d 6 HC = 3.75 cm.
Cting theo h&thirc luqng trong tam g i L vubng ABC, ta lai c 6 :
A B=
~ BH . BC = 2.4.6,15 = 14.76. d o d 6 AB = 3.84 (cm).
A C ~ = C HB
. C=3,75.6,15 =23,0625,dod6AC=4,8cm
TrB lbi : DQ d i i c i c canh g6c vubng cha tam g i i c l i :
AB .t 3.84 cm ; AC .t 4.8 cm.

1. Cho tam g i i c ABC vubng b A, dubng cao AH. Biet .AB : AC = ?. : I ,
AH = 42 c m . ' ~ i n hBH,'HC.
2. Cho tam g i i c ABC vuang b A, dubng cao AH. Bigt BH : HC = 9 : 16,
AH = 4 8 cm. Tinh do d i i c i c canh g6c vubng cua tam giic.
3. 'Trong met tam g i i c vuang, ti s 6 giCia dubng cao v i trung tuygn kk ttir
dinh g6d vubng bhng 40 : 41. Tinh do d i i c i c canh g6c vuang cua tam
g i i c vubng 66. bi6t canh huy@nbhng &
I
cm.
4. Cho hinh vubng ABCD v i diim 1 nitn giiia A v i B. Tia DI cat BC b E.
D u h g thing kk qua D vuang g6c viri DE c8t BC b F.
a) Tam g i i c DIF l i tam g i i c gi ? Vi sao ?

I
I
b) Chimg minh ring ---;-+khbng ddi khi I chuy4n dong tr&n
DI- DE'
doan AB.
5. Cho tam g i i c ABC vubng b A c6 canh AB = 6 cm, BC = 10 cm. CBc
d u h g phin g i i c trong v i ngoii cua g6c B cat AC Ian luut b D v i E.
Tinh c i c doan thing BD v i BE.
6. Cho tam g i i c ABC vubng b A, phin g i i c AD, dubng cao AH. Big1
CD = 68 cm, BD = 5 1 cm. Tinh BH, HC.
7. Cho tam g i i c nhon ABC, hai d u h g cao BD v i CE c i t nhau tai H.
Goi B I , C I l i hai dikm tuang irng tren c i c doan HB, HC. Bigt
AB,C = AC,B = 90'.

-~
~-~

Tam g i i c ABICl l i tam g i i c gi ? Vi sao ?
8. Canh huykn cha mot tam g i i c vubng 1611hun met canh g6c vubng c i ~ a
tam g i i c 1 i 9 cm, chn tdng hai canh g6c vubng 1131
hon canh huyen lh
6 cm. Tinh chu vi v i dien tich cua t a m g i i c vubng d6.

9. Chci tam gi5c ABC c d
AH = c', HC = b'.

;
90'

.d u h g cao BH. Dgf BC = a, CA =.b, AB = c,

I


1
Chimg minh ring a2 = b2

+ c2 = 2bc'.

10. ChotamgiicABCc6 B = 6 0 ° , A C = 1 3 c m v i B C - B A =

I

Tinh dQ dai cilc canh AB, BC.

11. Cho tam giilc ABC c6

> 90°, d u h g cao j 3 ~ Dst
.
BC = a, C
AC=c,AH=c',HC=b'.
Chimg minh ring a2 = b2 + c2 + 2bcq.

12. Cho tam giic ABC can ir B v i di6m D tren canh AC. Bi&t BDC = qO,
AD = 3 dm, DC = 8 dm. Tinh d&dii canh AB.
$2. ~f

-

SSLUQNG GIAC CUA GOC NHON

.

,A

d6i
k&
slna = -; cosa = -., .
huyin
huyen
8 .
d6i
,
ki
&a.
t g a =; cotga = kg
d6i
canh huyen
B
N6u hai g6c nhon a vi c6 sina = sinp . C
fhoac cosa = cosp, hocc ttga = tgp, ho#c
Hl1ih4
cotga = cotgp) thi a = p.
N6u hai g6c phu nhau thi sin g6c niy bang casin g6c kia v i tang g6c
niy bing catang g6c kia.
N6ua++=90°thi:
sina = cosp ; cosa = sin0 ;
tga = cotgp ; cotgp = tga.

+

Vi du 3. Cho tam gi6c ABC vuang ti A, d u h g cao AH. Bi6t AB = 7.5
cm;AH=6cm.
a) Tinh AC, BC ;
b) Tinh cosB, cosC.
GiJi
a ) Tam giilc ABH vuang ti H, the0 djnh li
Py-ta-go, ta.c6 :
BH2 = A B2 - A H 2 = 7.5 2 -62 = 20,25
.
suy ra BH = &Z = 4,5 (an).
B.
H
C
Tam. giic ABC vuOng 6 A, c6 AH I BC,
Hi~rlr5
theo he th~Icluqng trong tam giic vuang, ta c6 :

A

I

' , ,

,

45. Hai tam gidc vudng ABC v i A'B'C ddng dang viri n h a u ( i =
c6 hai dubng cao h. h' tuung irng thuec canh huyen a v i a'.
Chirrlg minh :
a) aa' = bb' + cc' ;
1
I
b)-=-+-.
hh' bb'

-

900).

1
cc'

46. Cho tam giic nhon ABC, ba dubng cao AH, BI, CK.
7

2

7

Chirng minh SHTK= (I - cos' A - cos B - C O S - C ) . S ~ ~ , ~

47. Cho tam giAc ABC vu8ng u A.
a) K t dubng cao AA'. ~ o Ei v i F the0 thir tu 18 hinh chi& cha didm A'
CE AC"

tren AC v i AB. Chirng minh - = - .
BF , 4 ' ~ ~
b) ~ h Do I i met didm tr&n canh BC ; M v i N Ian Iuqt I i hinh chi&u c i a
di6m D tren AB v i AC. Chirng minh DB. DC = MA. MB + NA. NC.
48. Tren met qua doi c 6 mat c l i thdp
. B
cao loom. Tir dinh B v i chin C c i a r - - - thip nhin diim A b chgn ddi duiri cic
g6c tuung h g bang 60' VA 30' (11 11 ).
Tinh chi& cao h ciia quH ddi.

49.

6

dB cao 920 m, tir met m6y bay
truc thHng nguiri ta nhin hai diem A
v i B cha hai d%u met
nh5ng g6c so vdi d h g nlm ngang
cha met d&t c i c g6c ldn luqt IP
a=37'v?iP=31°.

Hi~rhI J

A

Tinh chi& dhi AB c6a cdu.

50. Cho tam giic AMB vudng b M. Qua B k6 dubng thing d vu8ng g6c v6i
AB. Goi H v i K ldn luqt l i hinh chi& c c l diem M tr&n.dutingthing d
v i tr€n AB. Cho bi6t &IAB = a ( a 5 459 vi AB = 2a.

a) Tinh MA, MB, MH the0 a vh a ;
b) Tinh MH the0 a v i 2 a ;
C)

Chirng minh : cos2a = 1'- sin2 a , cos2a = 2cos 2 a -

1.


5 1. DINH NGHfA VA S u XAC DINH D U ~ N G
T R

~ N

T&phap cbc didm clch didm 0 cd djnh met khoang bang R kh6ng
d6i (R > 0) 11 d u h g trbn tam 0 c6 bbn kinh bang R.
D u h g kinh 11 day cung l6n nhgt c6a dubng trbn.
Qua ba di&mkh6ng thing hhng, bao gib ciing v8 duac met d u h g trbn
v i chi m6t m i th6i.
D u h g trbn di qua ba dinh A, B, C c i a tam gibc ABC goi l i d u h g
trbn ngoai ti6p tam gilc ABC. Khi 66 tam gibc ABC goi lh tam gibc n6i
ti6p d u h g trbn.
Cbch xbc djnh mot d u h g trbn :
- MQt dikm 0 cho trudc vh met s6 thuc duung R cho truirc xbc djnh
met dubng trbn tam 0 bbn kinh R.
- Ba didm kh6ng thing hhng x b d/nh met d u h g trbn di qua ba didm
66.
V i & 6. Cho tam gibc ABC vu6ng b B, AB = 8cm, BC = 6cm. Goi D 11
didm d6i ximg c6a didm B qua AC.
a) Chirng minh r&ngb6n didm A, B, C, D chng thuoc met dubng trbn.

b) Tinh bbn kinh cba d u h g trbn n6i trong
Gicii
a) Theo d& bhi, didm D d6i ximg v6i
didm B qua AC n6n DA = BA, DC = BC.
AADC = AABC (c.c.c),

- -

suy ra ADC = ABC = 90".
Goi 0 lh trung didm cba AC. Trong cbc
tam g i k vu6ng ABC v i ADC c6 BO v i DO
IP cbc d u h g trung tuy6n thuec canh huy&n
AC n6n BO = OA = OC, DO = OA = OC.
Suy r a : O A = O B = O C = O D .
V$y b6n didm A, B, C, D c h g thubc
Hinh 12
dutfng trbn tam 0 d u h g kinh AC.
b) Tam gihc ABC vu6ng b B, the0 djnh l i Py-ta-go, ta c6 :
AC 2 = AB2 + AC' = g2 +62 = 100, suy ra AC = LO cm


.
1

I'
I

1

.'

VAy bin kinh R c6a dubng trbn ( 0 ) lh : R = -AC = - . 10 = 5(cm)
2
2

.

V i rl!r 7. ~ h g6c
b xAy = 450 vh didm B tr&ntia Ax sao cho AB = 3cm.
a) Dqng dubng trbn ( 0 ) di qua A vh B sao cho tam 0 nhm tr&ntia Ayb) Tinh b6n kinh d u h g trbn ( 0 ) .

Gidi
a) Cich dung :
- D;mg dubng trung trqc d c6a.
doan AB, d c i t tia Ay b 0:
- Dung d u h g trbn t8m 0 , bin
kinh OA thi dubng trbn ( 0 ) chinh Ih
d u h g trbn phai d ~ p g .
Chrhry nrinh :
Vi digm 0 thu&c d u h g trung
x
truc d c i ~ adoan AB n&n OA = OB, do
d6 dubngetrbn t$m 0 di qua A v i B.
>C
Hun n3a di6m 0 thudc Ay n&nd u h g
Hinh 13
trbn ( 0 ) I i d u h g trbn can dqng.
Bltn lu&r :
LuBn luan dqng d t y c m&tdubng trbn tho6 m&ny&uC ~ cha
U d6 bhi.

b) Goi giao d i h citad.viri A 3 Ih H. Tam giic AOH 19 tam g i b vubng
can ir H, ta co :
,

!
3J5
V$y bsn kinh d u h g tr6n (0)1B : R ,= -cm

2

51. Cho tam giic d$u ABC, hai d~rimgcao BD, CE.
a) Chimg minh b6n di&mB, C, D, E chng thuec met d u h g trbn ;
b) Goi G 18 giao diim cha BD vh CE. Chirng minh b6n didm A, E, D, G
chng thu&c m6t d u h g trbn. Tinh bin kinh cba d t h g trbn niy, bi&'t tam
gi6c d&u ABC c6 canh bing 8cm.

-

TOAN NANG CAO HlNH 9 2


. . ;

52. Cho d u h g trbn ( 0 ) . day AB. V day BC vu6ng g6c viri ABI
a) Cheng minh AC I i dudng kinh cha d u h g trbn ( 0 ) .
b) Tinh bin kinh d u h g trbn (O), bi6t AB = 12cm. BC,= 5cm:
53. Cho hinh vubng ABCD. Tren c i c canh AB. BC, CD, DA My theo thir tu
AM
BN
CP QD

c i c didm M, N, P, Q sao cho -= -= - = MB NC PD Q A '
a) Chirng minh tit giic MNPQ l i hinh vu6ng ;
b) Goi 0 l i giao didm hai dudng chCo AC v i BD. C h h g minh ban d i d ~ n
M, N, P, Q cfing n i m tr&n mbt dubng trbn tam I i diim 0.
54. Cho d u h g trbn tam 0 bin kinh 4cm, c6 tam b goc toa do. H i y x i c d!nh
v/ tri c6a c i c diim A, B, C dbi v6i duirng trbn, b16t toa d o clic di6m :

B

A ( - 2 ; ) ;

6); C (26;-243).

55. Goi I v i K theo thir t$ I i c i c diem nim tr&n canh AB, AD cua hinh
vubng ABCD sao'cho A1 = AK. Dubng thing k ~ ?qua A vu6ng g6c vCli
DI b P, c L BC b Q.
C h h g minh nBm didm C, D, K, P, Q cilng thuec mot dubng !rbn.
56. Cho hinh vu6ng ABCD, 0 I i giao didm hai dubng chCo AC v i BD. Goi
M, N 1611IU@ I i trung didm c i a OB, CD.

-

a) C h h g minh AMN = 90°, tir d6 suy ra bb'n didm A, M, N, D c i ~ n g
thuqic mqit d u h g tr6n ;
b) So s i n h AN viri MD.
57. ~ h tam
o g i i c ABC can 6 A c 6 AB = IScm, dubng cao AH = 9cm.
Tinh b i n kinh d u h g trbn ngoai tisp tam giic ABC.
58. Cho hinh vusng ABCD canh bdng a. Goi M v i N I i hai diim tujr 9 tr&n
c i c canh AB v i AD sao cho chu vi tam giic AMN bing 2a. Goi H 18

hinh chi& cua didm C tren MN.
Chirng minh rang didm H lubn lu6n thuqic mqit d u h g trbn c 6 djnh khi
hai diim M, N chuybn dong tr&nc i c canh AB v i AD.
59. Cho dubng trbn ( 0 ; R) v i m6t di6m A nam trong d u h g trbn d 6 (A
khlic 0). Tim tap hqp trung didm M c i a doan thing AB khi didm B
chuyin dong tren dubng trbn (0).

Tam cua dubng trbn Is ihm d6i xirng c i a d u h g trbn 66.
Bgt ki dudng kinh n i o cting ,itruc dPji xitng c i a d u h g trbn.

I

I


D u h g kinh vudng g6c v&imet day cung thi di qua tnrng dikrn c6a day gy. I
D u h g kinh di qua trung didm cua met day (kh6ng 18 dubng kinh) thi
vuang g6c vdi day &y.
Trong met d u h g trbn :
- Hai day bang nhau thi clch d&utam'.
- Hai day cdch dku tam thi bang nhau.
-Day Ian han thi gln tam hun.
- Day g l n tam hun thi ldn hun.
Vi [/canh AB, AC, bik't R = 3cm vh khoang cich tir 0 den AB vh AC Iln luut 18

dl 1
2&cm vh -cm.
2

Gidi

K t OH IAB, OK IAC,ta c6
I

:

AH = BH = -AB nen AB = 2AH.
2
I
AK = CK = -AC n t n
2
Tam gidc AOH vudng
li Py-ta-go, ta c 6 :

A

AH^ = OA'

- OH^ =
=9-8= 1
Hi11h 14

suy ra AH = Icm.
Do 66 AB = 2AH = 2cm
Tam gilc AOK vudng 13K, the0 d/nh li Py-ta-go, ta c6 :

5
suy ra AK = -(cm) .
2

TrH ldi : AB = 2cm, AC = 5cm.


Bdi t@p
60. Cho d u h g trbn ( 0 ; R) hai dtly cung AB v i CD c&t nhau tai diim M
nim &n ngoii d u h g trbn.
, a) Chimg minh ring n€u AB = CD thi MA = MC ;
b) T r u h g hqp AB > CD. HHy so sinh khoing cich til M d&n trung diim
cha c i c dtly AB, CD.
61. Cho d u h g trbn ( 0 ) v i didm I nim b&n trong dubng trbn. Chirng minh
rhng trong c i c dtly di qua I thi dily vuBng g6c vdi 0 1 c6 dB dii nho nhst.
62. Cho nira d u h g trbn ( 0 ) dubng kAh AD. Tr&n nira dubng trbn l&y hai
di&mB v i C. Bi8t AB = BC = 2&cm, CD = 6cm. Tinh bin kinh d u h g
trbn.
63. Cho dubng trbn (0; R), d u h g kinh AB, dtly cung AC.
a) Cho bi&t khoang cich tir 0 d8n AC, BC l8n IU@ l i 6cm v i 8cm.
Tinh d& d i i c i c dily AC, BC v i bin kinh d u h g trbn ;
b) Tr&n tia d6i c i a tia CA Igy diim D sao cho CD = CA. Tim tap hqp
trong tilm G ctia tam giic ABC khi C chuydn dong tr&nd u h g trbn ( 0 ) .
64. Cho dubng trbn ( 0 ) d u h g kinh AD, dily cung AB. Qua B k6 day BC
vuBng g6c vdi AD.
Tinh bin kinh cba d u h g trbn bigt AB = IOcm, BC = 12cm.
65. Cho d u h g trbn ( 0 ) bin kinh 5cm. Hai dtly AB v i CD song song v6i nhau
v i c6 do dii l8n luqt bing 8cm v i 9.6cm. Tinh khoing cich giila hai dtly

,

'

I

1

I

6~.
66. Cho d u h g trbn ( 0 ; R), dubng kinh AB, day cuhg DE. Tia DE c6t AB
b C. Bi&t DOE = 90' v i OC = 3R.
a) Tinh d o d i i CD v i CE the0 R ;
b) Chimg minh CD.CE = CA.CB.
67. Cho d u h g trbn (0.;
R) v i mot diim M nim b8n trong dubng trbn.
a) Hgy n&ucich dqng dtly AB nhan didm M Iim trung dl& ;
b) Tinh d&d i i day AB n6i trong ctlu a bi€t R = 7,5cm, OM = 2, lcm.
68. Cho nira d u h g trbn ( 0 ; R). Hai dily cung AB vA CD song song viri nhau
c6 do d i i 18n luat l i 32cm vh 24cm v i khohng giaa hai dtly I i 4cm.
Tinh bin kinh dubng trbn.

I

I


Goi d l i khoHng cich tir tam 0 cha d u h g trbn ( 0 ; R) d&'nd u h g thing a
thi vj tri tuang d6i cha d u h g thing v i d u h g trbn duqc bidu thj the0 bHng sau :
Vi tri tuung d6i cha

'

S6didm
chung

H&thirc
gi3a d v i R

I. D u h g thing v i d u h g trbn tit nhau

2

d
2. Dubng thing v i d u h g t r b n ,ti&px6c nhau

1

d=R

d u h g th&ng v i duirng trbn

1

3. D u h g thing v i d u h g trbn kh6ng giao
nhau

/

O

I

d>R

I

Vi du 9 Cho doan thing AB v i trung di&m 0 cha AB. Tr&n tang mot
nira m+t phing bir AB vZ tia Ax, By vu6ng g6c vdi AB. Tren cBc tia Ax, By
Igy theo thir tu hai dikm C v i D sao cho COD = 90'. K6 OH ICD.
a) Chimg m h h h g H thu6c d u h g trbn t h 0 ;
b) XBc djnh vj tri tucmg d6i cha d u h g thing CD v&i d u h g trbn (0).
Gidi
a) Tia DO cht tia d6i cba tia Ax 13 E.
Tam giBc vu6ng AOE v i tam giic vu6ng
BOD c6 :

-

.

AOE = BOD (hai g6c d6i dinh).
Do 66 AAOE = ABOD (g.c.g),

suy ra OE = OD v i D i = lii.
Tam gi6c vu6ng OHD v i tam gi6c vu6ng
OAE c6 :

-

H = A = 900
O D = O E (chirng minh tren)

Hirth

IS

Dl= E (chimg minh tren).
Do 66 AOHD = AOAE (canh huy&n- g6c nhon), suy ra OH = OA.
Vi OA = OB m i OA = OH nen OA = OB = OH. Vgy di&m H thu6c
d u h g trbn tam 0 d u h g kinh AB.


'

b) Do H thuoc dubng trbn tam 0 bin kinh OA, mh C D IOH tai H n&n
khoing cdch tir didm 0 d&n CD bing bin kinh d u h g trbn ( 0 ) nghya l i d =
'
R. Vay d u h g thing CD ti6p xlic vdi d u h g trbn ( 0 ) tai didm H.

.

69. Cho dikm M cdch d u h g thing xy l I 6cm. VB dubng trbn (M ; IOcm).
a) C h h g minh ring d u h g trbn (M) c6 hai giao didm v6i d u h g thang xy.
b) Goi hai giao didm n6i tr&n II P vh Q. Tinh do d i i PQ.
70. Cho hinh thang ABCD.c6 = = 90°, AB = BC = lcm, AD = 2cm.

71.

72.
73.

74.

Chimg minh ring d u h g thing AC ti6p xlic vdi dubng trbn (D ; f i cm).
Cho hinh vuBng ABCD, tr€n d u h g chCo BD 18y didm I sao cho BI = BA.
Dubng thing k6 qua I vu8ng g6c vdi BD cit AD 6 E.
a) So sinh c i c doan thing AE, EI, ID ;
b) Xbc djnh vj tri tucmg d6i c6a d u h g thing BD vdi d u h g trbn (E ; EA).
Cho g6c nhon xOy. Dung dubng trbn bin kinh 3cm, ti6p x6c vdi canh
Ox v i c 6 tam n i m tr€n canh Oy.
Dqng d u h g trbn bin kinh 2cm, ti6p xuc v6i d u h g thing xy cho trubc
v i di qua diem A cich xy m8t khoHng 3cm.
Cho d u h g trbn (0; 15crn), AB lh day cung c6a d u h g trbn. Tim tap
hqp trung dikm Vc6a AB khi AB thay d8i trong dubng trong (0),biQ
AB = 24cm.

,

54. TIEP TUYEN CIIAD U ~ N GTRON
T ~ N HC H ~ T
HA1 TIEP TUYEN CAT NHAU

Ti6p tuy€'n c6a d u h g trbn 1I d u h g thing chi c6 met dikm chung vdi
d u h g trbn.
Ngu met d u h g thing I i tigp tuy6n c6a mot d u h g trbn thi n6 vu4ng
g6c v6i b i n kinh di qua ti6p dikm.
N6u mot dubng thing di qua mot dikm cba dubng trbn vh vuBng g6c
vdi bin kinh di qua dikm 66 thi d u h g thing 8y I& met ti6p tuy6n cba
d u h g trbn.
N6u hai tiEp tuy6n cc6 m&t duimg trbn cdt nhau tai m4t dikm thi :
- Didm 66 cich d&uhai ti6p didm.

- Tia k t tir diem d6 di qua t&m 118 tia phan giic cha g6c tao biri hai ti&p
tuy6n.


I'

- Tia k t ttr tam di qua di6m 66 lh tia phrln giic cba g6c tao biri hai bin

kinh di qua ti&p dikm.
D u h g trbn tigp xlic vdi ba canh cba tam giic goi Ih d u h g trbn noi
tigp tam giic, cbn tam giic goi lh ngoai tigp d ~ r h trbn.
g
.m Dubng trbn ti&p xlic viri mot canh c6a tam giic vh tigp xlic v8i phPn
kko dhi cGa hai canh kia goi 18 dubng trbn bing ti&ptam gidc.

1!

1

V i tlg 10. Cho dubng trbn (0 ; 5cm) vh didm M nhm b6n ngohi d u h g
trbn. Qua M k& hai ti&p tuygn MA vh MB vdi d u h g trbn (A vh B lh tigp
dikm).
k&tigp tuygn v6i d u h g trbn d t MA, MB
Ti3 di&m C tr6n cung i1h6
IPn luqt 6 . P vh Q. Cho bi&'t AM IBM.
a) Tir gi6c'MAOB 11 hinh gi ? Vi sao 7
b) Tinh chu vi tam giic MPQ ;

-

C)Tinh g6c POQ
.
.
.

Gidr
a) Theo giH thigt :
MA, MB 18 tigp tuy&n cha d~rbngtrbn ( 0 )
c3 A vh J3 n6n OA 1 AM vh OB 1BM, Go d6

,.

A = 90' vh

6 = 90'.
MA IMB n t n

.
;
n

G = 90°.

.

Tir giic AMBO c 6
=
=
= 90'
,

n&n 18 hinh chii nhgt. Hinh chii nhgt nhy lai
A
P
M
c6 OA = OB n&n18 hinh vu6ng.
Hinh 16
b) Theo tinh chgt hai ti6p tuygn cc6 mot
d u h g trbn c l t nhau, ta c6 :
MA = MB, PA = PC vh QB = QC.
Chu vi tam giic MPQ bhng :
MP+PQ+QM=MP+PC+CQ+QM
= (MP + PA) + (MQ + QC)
= MA + MB.
Vi tir giic AMBO 18 hinh vu6ng (crlu a) n6n MA = MB = OA = 5cm.
Vgy chu vi tam giilcA4PQ = 5cm + 5cm = 1Ocm.
c) Cang theo tinh chgt hai tigp tuygn cka mot d u h g x r b n c i t nhau tai
met dikm, ta I+ c6 :

-

OP lh phrln giilc c6a g6c AOC, OQ 18 phan giic cba g6c COB.


I -1
0
= -AOB = - .90 = 4
5'-

2
2

Vi d41 I I. Cho tam g i i c ABC ctin Zr A. VE d u h g trbn t i m D d u h g kinh
BC c k AC v i AB Iin l u g Zr E v i F. Goi H 18 giao didm cba BE v i CF.
C h h g minh ring :
a) B6n digm A, E. H, F tang thuec met d i r h g trbn.
b) DE 18 ti&p tuy&n c i a dubng trbn n6i trong ciu a.
Gicii
a) D 18 t i m dubng trbn d u h g kinh BC
n&n DB = DE = DC, d o d6 tam giic BEC

A

vu6ng b E suy ra AEB = 900 .
Tuong tu AFC = 90'.
Goi 0 l i trung di8m cha AH, ta c6 OE,
O F I&n I ~ o it i trung tuy&n thuec canh huy&n
AH cGa hai tam giic vu6ng AEH v i AFH n&n
OA = OH = O E = OF. Do 66 b6n di&mA, E,
H, F thucc dubng trbn (0).
b) Vi H l i giao digm hai d u h g cao BE
v i CF cba tam g i i c ABC, AD l i dubng trung
tuy&n thuQc canh d i y BC n&n AD I BC, d o
66 ba didm A, H, D thing hhng.

B

D

C

Hinh 17

Tam g i k BDE c i n b D vi c 6 DB = DE n&n b1 = e l .
A

Tam g i b EOH c i n b 0 vl c 6 OE = OH n&n E2 = H2.
A

Mi

A

= fi2n.511 E2 = G I , nhung HI

+ 61 = 90°,

suy rz
+ E2 = 90' hay O ~ =D 90' hay OE I DE.
DE vu6ng g6c v6i bin kinh OE tai E n&nDE is ti&ptuygn cha d u h g tmn (0).

-

75. Cho dubng trbn ( 0 ; 5cm). dubng kinh AB, ti&p tuy&n Bx. Goi C l i met
.
di&mtr&nd u h g trbr. sao cho BAC = 300 ,tla
AC cfd Bx b E.
a) C h h g minh BC' = AC . CE ;
b) Tinh d o d i i doan BE.

'

76. Cho nha d u h g trbn (0)d u h g kinh AB Qua C thuac nka d u h g trbn, k6
tiCp tuy6n xy cua nha dubng trbn. Goi M v l N Ian Iuut I i hinh chi& cba
dikm A vb diim B tren xy. Goi H lh than dubng vu611g g6c k6 tir C
xudng AB. Chimg minh :
a) C lh trung dikm cha MN ;
b) CH* = AM .BN.
77. Cho d u h g trbn ( 0 ) v l d u h g thhng d kh6ng giao nhau. Dung t16p tuy6n
c6a dubng trbn ( 0 ) sao cho ti6p tuy6n d6 song song viri d u h g thing d.
78. Cho d u h g trbn ( 0 ) v l di6m M n i m btn ngoli d u h g trbn. Qua M vO
hai ti+ tuy6n MA, MB vdi dubng trbn (0)trong d 6 A, B l i c i c ti6p
dikm sao cho AMB = 9
0'. Qua dikm C tren cung nhb G k 6 ti6p tuy&n
v6i d u h g trbn ( 0 ) cdt MA v i MB I&nIuut b P vh Q.
C h h g minh rhng :
1
+ MB) < PQ <-(MA
+ MB).
3
2
79. Cho nira d u h g trbn ( 0 ) d u h g kinh AB, hai ti6p tuy6n Ax, By. Trtn Ax,
By igy the0 thit tu hai dikm C v l D. Bi& AC + BD = CD. Chimg minh :
a) G6c COD = 90'.
b) D u h g theng AB lb ti6p tuy&n c6a d u h g trbn ngoai ti6p tam giic
COD, cbn d u h g theng CD 1b ti6p tuy&n c6a dubng trbn (0).
80. Cho g6c xAy k h i c g6c bet. Chimg minh ring c6 thk tim duuc v6 s b chc
dikm B vb C tr&n hai canh Ax v l Ay sao cho chu vi tam g l l c ABC lu6n
luBn bhng 2 1 (1 l l mijt dij dhi cho trudc).
81. Cho d u h g trbn (0 ;6cm) v l day AB bing IOcm. Goi M 18 mot didm tren
d u h g thang AB vh M nam btn ngobi d u h g trbn (0). Tim khoing cich

t& M d&n trung didm c6a AB khi g6c xen giiia hai ti&p tuy&n k6 tir M
bhng 60' ;90'.
82. Cho nira dubng trbn ( 0 ) d u h g kinh AB, ti6p tuy6n Ax. Qua dikm C trtn
nira d u h g trbn ( 0 ) k6 ti6p tuy6n vdi nira d u h g trbn cat Ax b M. K6
C H I AB cdt BM b I. C h h g minh I Ib trung dikm cba CH.
83. Cho tam g i i c ABC c6 BC = IOcm, CA = 12cm v l AB = 14cm. Tinh
khoing c i c h giira tam d u h g trbn niji ti+ vh t r y g tam cha tam giic.
84. Cho tam g i i c ABC c6 BC < AC, trung tuy&n CD. D u h g trbn n6i tiep
c8c tam giic ACD Vh BCD ti6p xxc vdi CD lln luqt b E vb F.
C h h g minh : 2EF = AC - BC.
85. Cho tam g i L ABC vu6ng ir A. D u h g trbn ( 0 ) n&i ti&p tam gi6c ti6p
x6c vdi canh AB, AC Iln luqt b D v l E.
a) Tit g i L ADOE l l hinh gi7 Vi sao?
b) Tinh b i n kinh d u h g trbn (O), biB AB = 5cm ; AC = 12cm.

-

MA

-


86. Cho tam gi5c ABC, bigt BC

= a, CA = b, AB = c. Goi r l i bin kinh
.,

dtrirng trbn noi tigp, S 18 dien tich c i ~ atam glac.
r(a + b + C)
Chirng minh S =

287. Dliaiig trbn noi tigp tam giic ABC ti$ xuc vdi canh BC tai D. Chirng
i tam giic ABC vuong 6 A I i :
rninh ring di&ukien c4n v i d ~dd
AB . AC = 2BD. DC.
88. Cho tam giic ABC vu6ng 6 A. Goi r, R the0 thir tu I i bin kinh duirng
trbn noi tiBp, ngoai tigp tam gi5c.
Chirng minh ring : AB + AC = 2(r + R).
89. Cho tam gidc ABC vu6ng b A c6 BC = a, CA = b, AB = c. Goi r lh bin
kinh d u h g trbn n6i tiBp tam giic.
r 45-1
Chirng minh ring : - 5 -.
a
2
90. Cho tam giic ABC c6 AB = 7,5cm, AC = 10,Scm va BC = 9cm. D u h g
trbn ( 0 , ) bing tiBp g6c A tiBp xuc v6i cach BC, tigp xuc vdi phSin kto
dii c6a hai canh AB, AC Iln luut 6 D, E, F.
Tinh do d i i c i c doan AE, AF, BE, CF.
91. Cho tam giic d&uABC. Goi M, N la hai di&mlSin luqt tr&nhai canh AB,
AC ; D 18 trung di6m c6a BC.
Bict chu vi tam g i k AMN blng nira chu vi tam gidc ABC. Tinh g6c MDN.
92. M6t tam giic vu6ng noi tiBp mot d u h g trbn duitng kinh 37dm v i ngoai
tigp met d u h g trbn b4n kinh 5dm. Tinh c i c canh g6c vu6ng cua tam
gihc do.

Goi 0,0' 18 tam cha hai duirng trbn. Dubng t h h g 00' goi 1 i d u h g
noi tam, doan thing 00' goi Ih doan n6i tam. D u h g n6i tam l i truc ddi
ximg c6a hinh g8m hai d u h g trbn ( 0 ) v i ('0') ;
NBu hai d u h g trbn ti6p xlIc nhau thi tiBp di&m nim tren d u h g noi
trlm.
NBu hai d u h g trbn cat nhau thi d u h g nbi tam vu6ng g k vdi drly

chung vh di qua trung dikm c6a drly chung.
Hai d u h g trbn ( 0 ; R) v i ( 0 ' ; r) c6 R 2 r. Vj tri tuang doi giira hai
d u h g trbn img vdi he thttc giira R, r v i 00' duuc cho the0 bHng sau :


He thlic gi3a 00'
vdi R vh r

S6 di6m
chung
2di'm
chung

Vi tri tuong dbi cOa hai d~rbng
trbn ( 0 ; R) vh (0'; r)

Hai d u h g trbn cit nhau

R-r<00'
Hai d u h g trbn ti6p xx6c nhau
- Ti&p x6c ngohi
1dih
00'=R+r
chung
00'=R-r
- Ti6p xx6c trong
Hai d u h g trbn khBng giao nhau
OO'>R+r
- d ngohi nhau

0 di6m
chung
OO'- d trone nhau
Tr&n hinh 18, c i c d u h g thing d l , d2 Ih ti+ tuy6n chung ngoii ciia
hai dubng trbn ( 0 ) v i (0'),c i c d u h g thing m,, m2 18 ti6p tuy6n chung
trong c6a hai dubng trbn ( 0 )v i (0').

I

,

I

'

/

I

(X
(I
yJ
d,-

'712

Hinh 18

1

V i dl( 12. Cho d u h g trbn ( 0 ) vh (0')tigp xuc ngohi tai A. Ti6p tuy6n
chung ngohi cba hai d u h g trbn c6 ti6p di6m vdi & h g trbn ( 0 ) 6 M vdi d u h g
trbn (0')6 N, ti&ptuy6n chung trong c6a hai d u h g trbn tai A cit MN 6 I.
a) Chirng minh tam giic MAN vh 010' 18 c i c tam giic vuBng ;
b) XBc djnh vj tri tumg d6i c6a d u h g thing MN vdi d u h g trbn dubng
kinh 00'.

Gidi
a) IM vh IA lh hai ti&p tuygn c6a
d u h g trbn (O), ta c6 IA = IM.
IN vh IA 1% hai tigp tuy6n c6a
d u h g trbn (a),
ta c 6 IA = IN.
Suy ra IM = IA = IN, do 66 tam
giic MAN lh tam giBc vu6ng 2, A.
?he0 tinh chgt hai ti6p ttuy&n c6a
Hinh 19
mQt d u h g trbn ckt nhau, ta lai c6 : I 0
vh 10' 1&nlugt lh tia phan g i k c6a hai
g6c k&bb MIA vh NIA ,do 66 I 0 1 10'. Vey tam giic 010' vuBng 2, I.
A


1

b) Goi I' 18 trung diim c l a 0 0 ' , .ta c6 1'1 = I 0 = 10' nen 1'1 18 bin kinh
d u h g trbn d u h g kinh 0 0 ' .
_ OM I MN vh O'N IMN ntn OM I/ O'N suy ra tir gihc OMNO' 18 hinh thang.
1'1 lh d u h g k n g binh c i a hinh thang OMNO nen I' 1 //OM, suy ra 1'1 I MN.

Dubng thing MN vuBng g6c v6i b6n kinh 1'1 tai I nen d u h g t h h g MN
lh t&p tuygn cc6 d u h g trbn (I).
V i (14( 13. Hai d u h g trbn (0,; 6.5cm) vh (02; 7,5cm) giao nhau tai A
vh B. Tinh dQ Cai doan n6i tam 0,02,bigt AB = 12cm.
~ i i i
'

Ta c6 O I O z 4 AB tai H.ntn HA = HB = 6cm.
Tam gi6c A 0 2 H vudng b H, ta

@
(1)

Hirrlr20

h)

O~H*
- AH^ = 7 5 *- 6*= 56.25 - 36 =,20,25 suy ra 02H = 4,5(cm).
Tam-gihc A O I H .vudng ir H, ta c6 :
2
2
2
2
O,H = O I A -AH = 6,5 - 62=2.2,25 - 36 =6,25 suy ra O I H = 2,5(cm).
- N6u o,',O2 thuoc hai nira mat phhng bir AB (h20a) thi :
0102
= 0 , H + H 0 2 = 2.5 + 4,5 = 7(cm).
- N 6 01,O2 thuQc mot nira
. .mat phang bir,AB (h20b) thi :

0,02= 0 2 H - O I H = 4,5 - 2,5 = 2(cm).
. .
Trd 1 8 :D o dhi doan n6i tam 0102
lh 7cm hoac 2cm.

~

,

Bbi top
93. Cho nira dubng trbn ( 0 ) d u h g kinh AB. VE d u h g trbn tam 0'd u h g kinh
b M.
OA. Qua A vi5 d&ycung AC &a d u h g trbn ( 0 ) c i t d u h g trbn (0')
C h h g minh :
a) D u h g trbn (0')
vh d u h g trbn ( 0 ) ti6p xxdc v6i nhau ;
b) O'M song song v6i OC ;
c) M 18 trung dikm c6a AC vh OM song song vdi BC.

.


94. Cho tam giic ABC vudng a A. VO d u h g trbn (01)
di qua A vh tiEp x6c
vdi BC tai B, vE d~rimgtrbn (02)
di qua A v i ti+ xxc vdi BC tai C. Goi
M 1 i trung didm cSa BC. Chdng midh :
a) D u h g trbn (0,)v i (02)
ti&$ xx6c vbi nhau ;
b) AM I i ti+ tuyEn chung c6a hai d u h g trbn (0,)v i to2).

95. Cho hai d u h g trbn (0 ; R ) v i (0' ; R') c i i nhau tai A v i B. BiEt
OAO'=~O",R = 6cm v i R' = 4,5cm.
a) Tinh 00, AB ;
b) Goi P I i trung didm c6a 00, qua A kc5 c i t tuy&n vudng g6c vbi AP
cit dubng trbn ( 0 ) if C, c&td u h g trbn (0')13D. So sdnh AC, AD v i AB.
96. Cho hai dubng trbn (01; 17cm) v i ( 0 2 ; 1Ocm). AB I i mot tigp tuygn
chung ngoii c6a hai d~rbngtrbn c6 ti6p didm viri d~rbngtrbn ( 0 , ) u A,
vdi d u h g trbn ( 0 2 ) if B. Dubng thing AB c i t dubng ndi tam 0101
b C.
= 21cm.
Tinh d o &hi c i c doan C O I , C 0 2 bi6t 0102
97. Cho td giic ABCD. Bi&i ring dubng trbn nOi tiCp hai tam giic ABC vh
ADC ti6p xx6c nhau. Chdng minh rgng d~rbngtrbn noi tiEp hai tam giic
ABD vh CBD c b g tiEp xx6c nhau.
98. Cho hai d u h g trbn (0,)v i (02)
ti6p xbc ngohi tgi A. ~ d dubng
t
thing
ti+ xuc vbi d u h g trbn (0,)ir B, ti+ xxlic v6i d u h g trbn (02)
if C. BiEt
AB = 6cm, AC = 8cm.
a) Tinh dQ dhi doan BC ;
b) Tinh bin kinh c6a c i c d u h g ( 0 , ) vh (0.9.
99. Cho hai dubng trbn ( 0 ) va (0') tiEp xhc ngohi u A. Dubng nbi t2m 00'
cit dubng trbn ( 0 ) b B, cit d u h g trbn (0') b C. DE 11 mot ti€$ tuy&n
chung ngohi cha hai dubng trbn (D E(O), EE(0')).
Goi M l i gino didm cua hai dubng thing BD v i CE. Chirng mich :

-

-

a) G6c EMD = 90' ;
b) MA lh tiEp tuy&n chung cda hai dubng trbn ( 0 ) v i ( 0 ' ) ;
c) MB.MD = ME.MC.
100. Cho hai d u h g trbn ( 0 ) v i (0')tiCp xx6c ngoii b A. Goi OM v i OM'11
cdc bin kinh c6a hai d u h g trbn v i OM // O'M'.
a) Chimg minh rang duirng thing MM' ludn lu6n di qua mot didm c 6
dinh S khi clic bin khi OM v i O'M' thay d6i ;
b) Tinh SO v i SO' b i d bin kinh hai d u h g trbn ( 0 ) v i (0')l&nIuat blng
5cm v i 3cm :