laisac.vecto

Nguyn Ph Khnh Lt http://www.toanthpt.net

1

CHNG I: VECT I. nh ngha:

1. Vect AB la mot oan thang co nh hng t A en B, k hieu ABJJJG

. a) A: iem goc. B b) B: iem ngon. c) ng thang AB: gia cua AB

JJJG. A

2. Phng cua ABJJJG

: tap hp cac ng thang song song vi ng thang AB, hoac trung vi ng thang AB.

3. Hng cua ABJJJG

: hng (chieu) t A en B theo phng cua ABJJJG

. 4. Moun cua AB

JJJG, k hieu AB

JJJG la o dai cua oan thang AB.

5. Vect khong, k hieu 0G

, la vect co moun bang 0 (iem goc va iem ngon trung nhau). 0G

co phng va hng tuy y. II. Vect cung phng, cung hng, ngc hng: B 1 2 3d // d // d (d1, d2, d3 cung phng) E D

a) ABJJJG

va CDJJJG

cung phng, cung hng. A F b) AB

JJJG va EFJJJG

cung phng ngc hng. C III. Vect bang nhau, oi nhau, t do:

1. Vect bang nhau:

AB va CD cung phng

AB CD AB va CD cung hng

AB CD

= =

JJJG JJJGJJJG JJJG JJJG JJJG

JJJJGJJJG

2. Vect oi nhau:

AB va EF cung phng

AB va EF oi nhau AB va EF ngc hng

AB EF

=

JJJG JJJGJJJG JJJG JJJG JJJG

JJJGJJJG

K hieu: AB EF; AB BA= = JJJG JJJG JJJG JJJG 3. Vect t do: la cac vect bang nhau a b c...= = vi goc tuy y.

IV. Phep cong va tr vect:

1. Tong cua hai vect: cG

a) nh ngha: OA a, AB b,OB c= = =JJJG G JJJG G JJJG G bG bG

Neu OB OA AB= +JJJG JJJG JJJG th c a b= +G G G . aG b) Quy tac ba iem cua phep cong vect:

O, A, B bat ky: OB OA AB OB OA AC CD DB= + = + + +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG aG


2

c) Tnh chat cua phep cong vect:

( ) ( )a b b a

a b c a b c

+ = ++ + = + +

G G G GG G G G G G ( )

a 0 0 a a

a a 0

+ = + =+ =

G G G G GG G G

d) Quy tac hnh bnh hanh: A C

OA OB OCOACB la hnh bnh hanh

+ =JJJG JJJG JJJG 2. Hieu cua hai vect: O B

a) nh ngha: ( )a b a b = + G G G G A b) Quy tac ba iem cua phep tr vect:

O, A, B bat ky: OB OA AB =JJJG JJJG JJJG O B V. Phep nhan vect:

1. nh ngha: Cho a 0, m R, m 0 G G

b cung hng vi a neu m>0

ma b : b ngc hng vi a neu m


3

nen: AB DC FEGH DC

gt : FG EH FE GH

= = == =

JJJG JJJG JJJG JJJG JJJGJJJG JJJG JJJG JJJG A D

Do G, H, D, C khong thang hang. Vay CDGH la hnh bnh hanh. B C

2. Cho bon iem A, B, C, D. Tnh cac vect sau: a) v AB DC BD CA= + + +G JJJG JJJG JJJG JJJG . b) u AB CD BC DA= + + +JG JJJG JJJG JJJG JJJG .

Hng dan:

a) ( ) ( )v AB DC BD CA AB BD DC CA AD DA 0= + + + = + + + = + =G JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG G . b) ( ) ( )v AB CD BC DA AB BC CD DA AC CA 0= + + + = + + + = + =G JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG G .

3. Cho bon iem A, B, C, D. Chng minh rang AB CD AC DB = +JJJG JJJG JJJG JJJG . Hng dan: AB CD AC DB AB CD AC BD AB BD AC CD AD AD = + = + = + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG (ang thc ung). Cach khac: ( ) ( )AB CD AC CB CB BD AC BD AC DB = + + = = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG .

4. Cho hnh bnh hanh ABCD, O la giao iem hai ng cheo. Chng minh OA OB OC OD 0+ + + =JJJG JJJG JJJG JJJG G . Hng dan: O la giao iem cua hai ng cheo AC va BD: ( ) ( )OA OB OC OD OA OC OB OD 0+ + + = + + + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG G

5. Cho sau iem A, B, C, D, E, F. Chng minh: AD BE CF AE BF CD+ + = + +JJJG JJJG JJJG JJJG JJJG JJJG . Hng dan:

( ) ( ) ( )( ) ( )

AD BE CF AE ED BF FE CD DF

AE BF CD ED DF FE

AE BF CD

+ + = + + + + += + + + + += + +

JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJGJJJG JJJG JJJG JJJG JJJG JJJGJJJG JJJG JJJG

6. Cho hai vect ( )a va b a, b 0G G . Hay tm moi quan he gia a va bG G neu co mot trong hai ieu kien

sau: a b a b+ = +JG JJGG G ; a b a b+ = G G G G . A Hng dan: a

G b

G

Neu a b a b+ = +JG JJGG G O c a b= +G G G B Ta co: OB OA AB A nam gia O va B a, b cung hng= + G G


4

Neu a b a b+ = G G G G A bG B Ta co: OB CA= aG Hnh bnh hanh OABC co hai ng cheo bang nhau OABC la hnh ch nhat OA OC a b G G O C

7. Cho t giac ABCD, I va J lan lt la trung iem cua hai ng cheo AC va BD. Chng minh AB CD 2IJ+ =JJJG JJJG JJG . Hng dan: A ( ) ( )AB CD AI IJ JB CI IJ JD+ = + + + + +JJJG JJJG JJG JJG JJG JJG JJG JJG ( ) ( )2IJ AI CI JB JD= + + + +JJG JJG JJG JJG JJG B D 2IJ= JJG C

8. Cho tam giac ABC. Goi M, N, P lan lt la trung iem cua BC, CA, AB. Chng minh: AM BN CP 0+ + =JJJJG JJJG JJJG G . Hng dan:

( ) ( ) ( ) 1 1AM BN CP AB BM BC CN CB BP AB BC MC CN AB MN 02 2+ + = + + + + + = + + + = + =JJJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG G . V 1 1BC CB 0, BP BA, BM MC, MN BA

2 2+ = = = =JJJG JJJG G JJJG JJJG JJJG JJJG JJJG JJJG

Cach khac: Dung quy tac trung iem

( )( )( )

1AM AB AC2

1BN BA BC pcm21CP CA CB2

= + = + = +

JJJJG JJJG JJJG

JJJG JJJG JJJG

JJJG JJJG JJJG

9. Cho t giac ABCD, M va N lan lt la trung iem cua AD va BC.

Chng minh rang ( )1MN AB DC2= +JJJG JJJG JJJG . Hng dan:

( ) ( )( ) ( ) ( )

1 1MN MB MC MA AB MD DC2 21 1 1 MA MD AB DC AB DC2 2 2

= + = + + +

= + + + = +

JJJG JJJG JJJG JJJG JJJG JJJG JJJG

JJJG JJJG JJJG JJJG JJJG JJJG


5

10. Cho hai vect a va bG G

. Chng minh rang: a) a b a b+ = +G G G G b) a b a b+ = G G G G

Khi nao xay ra dau ang thc? Hng dan:

a) Dng OA a, AB b,OB a b= = = +JJJG G JJJG G JJJG G G . Vi ba iem O, A, B luon co OB OA+AB hay a b a b + +G G G G . Dau " " xay r a khi O, A, B thang hang va A nam tr ong OB= .

b) Dng OA a, OB b. Ta co: a b OB OA BA= = = =JJJG G JJJG G G G JJJG JJJG JJJG . Suy r a: a b AB AB OA OB a b = = = G G JJJG G G . Dau " " xay r a khi a // b= G G .

11. Cho oan thang AB va hai so , khong ong thi bang 0. Chng minh rang: a) Neu 0 th ton tai duy nhat iem M sao cho MA MB 0 + + =JJJG JJJG G . b) Neu 0 th khong ton tai iem M sao cho MA MB 0 + = + =JJJG JJJG G . c) Neu 0 th v MA MB khong oi, khong phu thuoc v tr iem M + = = + G JJJG JJJG . d) ( )Neu 0 th vi moi iem M, ta co: MA MB MI, + + = + JJJG JJJG JJJG trong o I la iem xac nh

bi IA IB 0 + =JJG JJG G . e) Neu 0, M va N xac nh bi MN MA MB + = + JJJG JJJG JJJG .

Chng minh rang ng thang i qua mot iem co nh. Hng dan:

a) ( ) ( )MA MB 0 MA AB AM 0 AM AB AM AB + = + = + = = +JJJG JJJG G JJJG JJJG JJJJG G JJJJG JJJG JJJJG JJJG ton tai duy nhat M .

b) Gia s M sao cho ( )MA MB 0 MA MB 0 MA MB 0 + = = =JJJG JJJG G JJJG JJJG G JJJG JJJG G BA 0 =JJJG G 0 0 = = : trai gia thiet. Vay khong ton tai M thoa yeu cau bai toan.

c) v MA MB BA la vect khong oi.= + = G JJJG JJJG JJJG d) ( ) ( ) ( ) ( )MA MB MI IA MI IB MI IA IB + = + + + = + + +JJJG JJJG JJJG JJG JJJG JJG JJJG JJG JJG

Vay ( )MA MB MI hay MI MA MB + = + = + + +JJJG JJJG JJJG JJJG JJJG JJJG

.

e) at ( )MN MA MB MN MI MN // MI M, N, I thang hang= + = + JJJG JJJG JJJG JJJG JJJG JJJG JJJG Vay ng thang MN luon qua iem I co nh.

12. Cho tam giac ABC. Goi A1, B1, C1 la cac iem xac nh bi 1 12A B 3A C 0+ =JJJJG JJJJG G

, 1 12B C 3B A 0+ =JJJJG JJJJG G

. Chng minh rang tam giac ABC va A1B1C1 co cung trong tam.


6

Hng dan: Gia thiet ta co:

( ) ( )11 1 1 1 1 11

2GB 3GC 5GA

2GC 3GA 5GB 5 GA GB GC 5 GC GA GB GG 0 hay G G

2GA 3GB 5GC

+ = + = + + = + + = + =

JJJG JJJG JJJJGJJJG JJJG JJJJG JJJG JJJG JJJG JJJJG JJJJG JJJJG JJJJG GJJJG JJJG JJJJG .

Trong o G la trong tam tam giac ABC, G1 la trong tam tam giac A1B1C1.

13. Cho hai vect a, b khac 0G G G

va khong cung phng. Goi u, vJG G

la hai vect nh bi 1 1u a b= + JG G G

,

2 2v a b= + G G G

. Chng minh rang 1 2 1 2u v va = = = JG G

, con u, vJG G

cung phng 1 2 2 1 0 = . Hng dan:

( ) ( )1 1 2 2 1 2 2 1u v a b a b a b= + = + = JG G G G G G G G (1) ieu nay vo ly neu 1 2 2 10 hoac 0 . Vay ( ) 1 2 2 1 1 2 1 21 0 va = = = = .

Ta co 2 21 2 1 1u va v cung phng k ,k R; k k 0 + >JG G

Sao cho ( ) ( ) 1 1 2 21 2 1 1 2 2 1 1 2 21 1 2 2

k k 0k u k v 0 k k a k k b 0

k k 0 + =+ = + + + = + =

JG G G G G G.

He co nghiem khi 1 2k k 0= = ieu kien 1 22 21 2 1 2 2 1

1 2

k k 0 D 0 0

+ > = = =

14. A, B, C la ba iem phan biet.

Chng minh rang: A, B, C thang hang AB va AC cung phng JJJG JJJG . Hng dan: Thuan: A, B,C thang hang AB va AC cung gia AB, AC cung phng JJJG JJJG JJJG JJJG . ao: Neu AB, AC

JJJG JJJG cung phng th hai ng thang AB, AC cung phng. Nhng hai ng thang nay

co chung iem A nen trung nhau. Suy ra A, B, C thang hang

15. Cho hnh thang ABCD co hai ay AB, CD vi AB 2CD= . T C ve CI DA=JJG JJJG . Chng to: a) I la trung iem AB. b) DI CB=JJG JJJG .

Hng dan:

a) Do CI DA=JJG JJJG nen CIAD la hnh bnh hanh AI // CD . Do o I tren AB.


7

Mat khac: AI DC AB

AIAB 2DC 2

= = = I la trung iem AB.

b) CIAD la hnh bnh hanh DC AI

DC IB DCIB la hnh bnh hanh DI CB I la tr ung iem AB nen AI IB

= = ==

JJJG JJG JJJG JJG JJG JJJGJJG JJG

16. Cho hai hnh bnh hanh ABCD va ACEF.

a) Dng cac iem M, N sao cho EM BD, FN BD= =JJJG JJJG JJJG JJJG . b) Chng minh rang CD MN=JJJG JJJG .

Hng dan:

ABCD la hnh bnh hanh CD BA

CD EFABEF la hnh bnh hanh EF BA

= = =

JJJG JJJG JJJG JJJGJJJG JJJG

EM BD

EM FN EMNF la hnh bnh hanh MN EFFN BD

= = ==

JJJG JJJG JJJG JJJG JJJG JJJGJJJG JJJG

17. Cho hnh bnh hanh ABCD. Dng cac iem M, N thoa man:

a) MA MB MC AD =JJJG JJJG JJJG JJJG . b) NC ND NA AB AD AC+ = + JJJG JJJG JJJG JJJG JJJG JJJG . c) Chng minh MN BA=JJJG JJJG .

Hng dan:

a) MA MB MC AD BA MC AD CM AD BA AD AB AC = = = = + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG C la tr ung iem AM

b) AC ND AC AC DN AC N la nh th t cua hnh bnh hanh DACN+ = = JJJG JJJG JJJG JJJG JJJG JJJG . c) T cau a va b CM DN DCMN la hnh bnh hanh CD MN = =JJJG JJJG JJJG JJJG

Tng t BA CD MN BA= =JJJG JJJG JJJG JJJG .

18. Cho tam giac eu ABC canh a. Xac nh vect AB AC+JJJG JJJG va tnh moun vect nay. Hng dan: Ve trung tuyen AM, keo dai AM lay iem E sao cho ME AM= . Ta co: AB AC 2AM AE+ = =JJJG JJJG JJJJG JJJG . Do o: a 3AB AC 2 AM 2. a 3

2+ = = =JJJG JJJG JJJJG .

19. Cho hnh vuong ABCD canh a. Xac nh vect ( )1 AB AC AD2 + +JJJG JJJG JJJG va tnh moun vect nay.


8

Hng dan:

( ) ( ) ( )1 1 1AB AC AD AB AD AC AC AC AC2 2 2 + + = + + = + = JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG Do o: ( )1 AB AC AD AC a 22 + + = =JJJG JJJG JJJG JJJG .

20. Cho tam giac eu ABC canh a, trc tam H. Tnh moun HA, HB, HCJJJG JJJG JJJG

. Hng dan:

2 2 a 3 a 3HA HB HC AA .3 3 2 3

= = = = =JJJG JJJG JJJG (AA la ng cao).

21. Cho hnh vuong ABCD tam O canh a. Xac nh moun cac AB AD, AB AC, AB AD+ + JJJG JJJG JJJG JJJG JJJG JJJG Hng dan: Theo quy tac hnh bnh hanh: AB AD AC AB AD AC AC a 2+ = + = = =JJJG JJJG JJJG JJJG JJJG JJJG . Ve CA AB. Ta co: AB AC AC CA AA = + = + =JJJG JJJG JJJG JJJG JJJG JJJG JJJJG

2 2 2 2AB AC AA AA AD DA a 4a a 5 (pitago) + = = = + = + =JJJG JJJG JJJJG . AB AD DA AB DB AB AD DB DB a 2 = + = = = =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG .

VAN E 1: CHNG MINH ANG THC VECT

9 Dung quy tac ba iem AB BC AC; AC AB BC+ = =JJJG JJJG JJJG JJJG JJJG JJJG . 9 M rong quy tac ba iem 1 2 2 3 n 1 n 1 nA A A A ... A A A A+ + + =

JJJJJG JJJJJG JJJJJJJG JJJJJG.

9 Quy tac rut gon: Neu 1 1 2 2 n nIA IA ... IA 0 + + + =

JJJG JJJG JJJG G th ( )1 1 2 2 n n 1 2 nMA MA ... MA ... MI + + + = + + + JJJJG JJJJJG JJJJJG JJJG .

9 Neu G la trong tam cua ( )1 2 nA , A , ..., A va G la trong tam cua ( )1 2 nB , B , ..., B th ta co 1 1 2 2 n nA B A B ... A B nGG+ + + =JJJJJG JJJJJG JJJJJG JJJJG

.

9 Quy tac hnh bnh hanh: Cho hai vect AB,CD khac 0JJJG JJJG G

va khong cung phng. Dng hnh bnh hanh ABCD. Ta co: AC AB AD 2AM= + =JJJG JJJG JJJG JJJJG .

BAI TAP

1. Cho t giac ABCD. M, N lan lt la trung iem AD, BC, O la trung iem MN. Chng minh: a) AD CD AC DB = +JJJG JJJG JJJG JJJG . b) ( ) ( )1 1MN AB DC AD BC BD2 2= + = + +JJJG JJJG JJJG JJJG JJJG JJJG . c) OA OB OC OD 0+ + + =JJJG JJJG JJJG JJJG G .


9

d) MA MB MC MD 4MO; M+ + + = JJJG JJJG JJJG JJJG JJJG e) Goi F la trung iem CD. Chng minh rang: ( )2 AB AN FA DA 3DB+ + + =JJJG JJJG JJJG JJJG JJJG .

Hng dan:

a) Quy tac ba iem: AB CD AC CB CD AC DC CB AC DB = + = + + = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG . b) Quy tac ba iem, trung iem: ( ) ( ) ( ) ( )AB DC AM MN NB DM MN NC AM DM NB NC 2MN+ = + + + + + = + + + =JJJG JJJG JJJJG JJJG JJJG JJJJG JJJG JJJG JJJJG JJJJG JJJG JJJG JJJG (2 vect oi nhau). c) Quy tac trung tuyen, trung iem: ( ) ( ) ( )OA OB OC OD OA OD OB OC 2OM 2ON 2 OM ON 0+ + + = + + + = + = + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG G d) ( ) ( ) ( ) ( )MA MB MC MD MO OA MB OB MC OC MD OD+ + + = + + + + + + +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG

4MO OA OB OC OD 4MO 0 4MO= + + + + = + =JJJG JJJG JJJG JJJG JJJG JJJG G JJJG e) ( ) ( )2 AB AN FA DA 3DB 2 DA AB FA AN 3DB+ + + = + + + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG

( )2 DB FN 3DB + =JJJG JJJG JJJG 2FN DB =JJJG JJJG : hien nhien ung.

2. Cho tam giac ABC va tam giac ABC co trong tam la G va G. a) Chng minh GA GB GC 0+ + =JJJG JJJG JJJG G . b) Chng minh AA BB CC 3GG + + =JJJJG JJJG JJJG JJJJG . c) Suy ra ieu kien can va u e hai tam giac co chung trong tam la AA BB CC 0 + + =JJJJG JJJG JJJG G . d) Goi G1, G2, G3 la trong tam BAC , CAB , ABC . Chng minh G la trong tam 1 2 3G G G . Biet

ABC va A B C co cung trong tam G. A Hng dan:

a) GB GC 2GM+ =JJJG JJJG JJJJG (tnh chat trung iem) N Ma AG 2GM=JJJG JJJJG G Nen GA GB GC AG 2GM 0+ + = + =JJJG JJJG JJJG JJJG JJJJG G

b) Quy tac ba iem: B M C

AA AG GG G A

BB BG GG G B

CC CG GG G C

= + + = + + = + +

JJJJG JJJG JJJJG JJJJGJJJG JJJG JJJJG JJJJGJJJG JJJG JJJJG JJJJG

( ) ( )AA BB CC 3GG AG BG CG G A G B G C + + = + + + + + +JJJJG JJJG JJJG JJJJG JJJG JJJG JJJG JJJJG JJJJG JJJJG Ma AG BG CG 0 (cau 1)+ + =JJJG JJJG JJJG G A G B G C 0 (tnh chat tr ong tam) + + =JJJJG JJJJG JJJJG G Nen AA BB CC 3GG + + =JJJJG JJJG JJJG JJJJG

c) G G GG 0 AA BB CC 0 = + + =JJJJG G JJJJG JJJG JJJG G


10

d) Theo tren ta co: 1 2 3AG BG CG 0+ + =JJJJG JJJJG JJJJG G

Ta co: 1AB AC AA 3AG+ + =JJJG JJJG JJJJG JJJJG

(G1 la trong tam tam giac BCA)

2BA BC BB 3BG+ + =JJJG JJJG JJJG JJJJG

(G2 la trong tam tam giac CAB)

3CA CB CC 3CG+ + =JJJG JJJG JJJG JJJJG

(G3 la trong tam tam giac ABC)

( )1 2 3AA BB CC AC BC CB AB BA CA 3 AG BG CG + + + + + + + + = + +JJJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJJG JJJJG JJJJG (1) Ma AA BB CC 0 + + =JJJJG JJJG JJJG G va ABC, A B C co chung trong tam G. Suy ra (1): ( )1 2 33 AG BG CG 0+ + =JJJJG JJJJG JJJJG G . Vay G la trong tam tam giac G1G2G3.

3. Cho hnh bnh hanh ABCD. a) Cho AB a, AD b= =JJJG G JJJG G , I la trung iem CD, G la trong tam tam giac BCD.

Chng minh rang 1BI b a2

= JJG G G , tnh AG theo a, bJJJG G G .

b) Neu G la trong tam tam giac BCI. Chng minh 5 2AG a b6 3

= +JJJJG G G . c) Tren ABC , goi A1, B1, C1 la cac iem xac nh bi 1 12A B 3A C 0+ =

JJJJG JJJJG G, 1 12B C 3B A 0+ =JJJJG JJJJG G

,

1 12C A 3C B 0+ =JJJJG JJJJG G

. Chng minh rang 1 1 1ABC va A B C co cung trong tam. d) Neu B1, C1 cau c la trung iem cua CA, AB. at 1 1BB u,CC v= =

JJJJG JG JJJJG G.

Tnh BC,CA, ABJJJG JJJG JJJG

theo u, vJG G

. Hng dan:

a) ( ) ( ) ( ) ( )1 1 1 1 1BI BC BD AD AD AB 2AD AB 2b a b a2 2 2 2 2 = + = + = = = JJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG G G G G 2 2 1AG AB BG AB BI a b a

3 3 2 = + = + = +

JJJG JJJG JJJG JJJG JJG G G G D I C

Vay 2 1AG a b a3 2 = +

JJJG G G G G

b) ( )3AG AB AC AI AB AB AD AI = + + = + + +JJJJG JJJG JJJG JJG JJJG JJJG JJJG JJG ( )2AB AD AD DI= + + +JJJG JJJG JJJG JJG A B 1 52AB 2AD AB AB 2AD

2 2= + + = +JJJG JJJG JJJG JJJG JJJG

Vay 5 2AG a b6 3

= +JJJJG G G (pcm) c) Goi G va G1 lan lt la trong tam 1 1 1ABC va A B C .

Ta co: ( ) ( )1 1 1 1 12A B 3A C 0 2 A G GB 3 A G GC 0 2GB 3GC 5GA+ = + + + = + =JJJJG JJJJG G JJJJG JJJG JJJJG JJJG G JJJG JJJG JJJJG . Tng t ta co: 1 1 12B C 3B A 0 2GC 3GA 5GB+ = + =

JJJJG JJJJG G JJJG JJJG JJJJG


11

1 1 12C A 3C B 0 2GA 3GB 5GC+ = + =JJJJG JJJJG G JJJG JJJG JJJJG

Vay t ba ang thc suy ra ( ) ( )1 1 15 GA GB GC 5 GA GB GC+ + = + +JJJG JJJG JJJG JJJJG JJJJG JJJJG

1 1 1

1 1 1 1 1 1 1 1 1

1 1

GA GB GC 0

GG G A GG G B GG G C 0

3GG 0 G G

+ + = + + + + + = =

JJJJG JJJJG JJJJG GJJJJG JJJJJG JJJJG JJJJJG JJJJG JJJJJG GJJJJG G

d) Ta co: 1BA BC 2BB 2u+ = =JJJG JJJG JJJJG JG

1CA CB 2CC 2v+ = =JJJG JJJG JJJJG G

hay CB BA CB 2CB BA 2v+ + = + =JJJG JJJG JJJG JJJG JJJG G

Vay: ( )( ) ( )

2BC u v2u BA BC 23 CA CB BA 2v u

2 32v BA 2BC BA 2u v3

= = + = + = + = = +

JJJG JG GJG JJJG JJJG JJJG JJJG JJJG G JGG JJJG JJJG JJJG JG G

4. Cho tam giac ABC, trong tam G.

a) Goi A1, B1, C1 lan lt la trung iem cua BC, AC, AB. at 1 1BB u,CC v= =JJJJG JG JJJJG G

. Chng minh rang

1 1 1AA BB CC 0+ + =JJJJG JJJJG JJJJG G

va tnh BC,CA, AB theo u va vJJJG JJJG JJJG JG G

. b) Goi I la iem tren canh BC sao cho 2CI 3BI= , F la iem tren canh BC keo dai sao cho

5FB 2FC= . Tnh AI, AFJJG JJJG theo AB va ACJJJG JJJG , tnh AG theo AI va AFJJJG JJG JJJG . c) H la iem oi xng cua B qua G. Chng minh ( )2 1 1AH AC AB; CH AB AC3 3 3= = +JJJG JJJG JJJG JJJG JJJG JJJG .

Hng dan:

a) Ta co:

( ) ( ) ( )11 1 1 11

AB AC 2AA

BA BC 2BB AA BB CC AB BA CA AC BC CB 0

CA CB 2CC

+ = + = + + = + + + + + =+ =

JJJG JJJG JJJJGJJJG JJJG JJJJG JJJJG JJJJG JJJJG JJJG JJJG JJJG JJJG JJJG JJJG GJJJG JJJG JJJJG

BA BC 2u

Theo tr en:CA CB 2CB BA 2v

+ = + = + =

JJJG JJJG JGJJJG JJJG JJJG JJJG G

( ) ( ) ( )2 2 2BC u v ; BA 2u v AB 2u v3 3 3= = + = +JJJG JG G JJJG JG G JJJG JG G

b) Ta co: ( ) ( )( ) ( )

2 CA AI 3 BA AI 02CI 3BI 05BF 2CF 0 5 BA AF 2 CA AF 0

+ + + = + = = + + =

JJJG JJG JJJG JJG GJJG JJG GJJJG JJJG JJJG JJJG G

3 2 3 25AI 3AB 2AC AI AB AC va AF AB AC5 5 5 5

= + = + = JJG JJJG JJJG JJG JJJG JJJG JJJG JJJG JJJG Ta lai co: 3AG AA AB AC AB AC= + + = +JJJG JJJG JJJG JJJG JJJG JJJG


12

Do

3 2 5 3AF AB AC AB AI AF

5 5 8 83 2 25 9AI AB AC AC AI AF5 5 16 16

= = + = + =

JJJG JJJG JJJG JJJG JJG JJJG

JJG JJJG JJJG JJJG JJG JJJG

Vay 1 1 35 1

AG AB AC AI AF3 3 48 16

= + = JJJG JJJG JJJG JJG JJJG .

c) ( )2 2 1AH AB 2AG AA AB AC hay AH AC AB(pcm)3 3 3+ = = + + = JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG ( )2 1 2 1 1 1CH CA CB CA CA AB CA AB3 3 3 3 3 3= = + = JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG Vay 1 1CH AC AB

3 3= JJJG JJJG JJJG .

VAN E 2: XAC NH MOT IEM THOA MAN MOT HE THC VECT Bai 1: Cho tam giac ABC, hay dng cac iem I, J, K, L biet rang: A L

1) IA 2IB 0 =JJG JJG G . 2) JA JB 2JC 0 =JJG JJG JJG G . 3) KA KB KC BC+ + =JJJG JJJG JJJG JJJG . K G 4) 2LA LB 3LC AB AC + = +JJJG JJJG JJJG JJJG JJJG

Hng dan: B C 1) ( )IA 2IB 0 IA 2 AB AI 0 AI 2AB = = =JJG JJG G JJG JJJG JJG G JJG JJJG .

Vay I la iem oi xng cua A qua B. J

2) 1JA JB 2JC 0 BA 2CJ 0 CJ AB2

= + = =JJG JJG JJG G JJJG JJG G JJG JJJG 3) Cach 1: Goi G la trong tam tam giac ABC, ta co: I

KA KB KC 3KG+ + =JJJG JJJG JJJG JJJG Vay 13KG BC GK CB

3= =JJJG JJJG JJJG JJJG , K thuoc AB.

Cach 2: Ta co: BC KC KB= JJJG JJJG JJJG Vay ( ) 2KA KB KC KC KB KA 2KB 0 KA 2 AB AK 0 AK AB3+ + = + = + = =JJJG JJJG JJJG JJJG JJJG JJJG JJJG G JJJG JJJG JJJG G JJJG JJJG Vay K thuoc AB.

4) Ta co: ( ) ( )2LA LB 3LC AB AC 2AL AB AI 3 AC AL AB AC + = + + = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJG JJJG JJJG JJJG JJJG 14AL 2AC 2AB AL BC

2 = =JJJG JJJG JJJG JJJG JJJG .

Bai 2: Cho tam giac ABC. Hay dng iem I, F, K, L sao cho:

1) 2IA 3IB 3BC =JJG JJG JJJG . 2) FA FB 2FC 0+ + =JJJG JJG JJJG G .


13

3) 2KA KB 2CB CA+ = +JJJG JJJG JJJG JJJG . 4) IA IB IC BC+ =JJG JJG JJG JJJG . 5) FA FB FC AB AC+ + = +JJJG JJG JJJG JJJG JJJG . 6) 3KA KB KC 0+ + =JJJG JJJG JJJG G . 7) 3LA 2LB LC 0 + =JJJG JJJG JJJG G .

Hng dan: 1) ( )2AI 3 IB BC 3IC 3AC IA 0= + = + =JJG JJG JJJG JJG JJJG JJG G .

Vay I la iem tren ACJJJG

sao cho AI 3AC=JJG JJJG . 2) Goi M la iem gia oan AB, ta co: FA FB 2FC 0 2FM 2FC 0 FM FC 0+ + = + = + =JJJG JJG JJJG G JJJG JJJG G JJJG JJJG G .

Vay F la iem gia cua oan MC. 3) Goi N la iem tren BA sao cho 2NB NA 0+ =JJJG JJJG G va P la iem tren BA sao cho 2PA PB 0+ =JJJG JJJG G .

Ta co: BA

2KA KB 2CB CA 3KP 3CN hay KP CN CK NP3

+ = + = = = =JJJGJJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG

Vay K thoa 1CK BA3

=JJJG JJJG . 4) Ta co: IA CB BC IA 2BC+ = =JJG JJJG JJJG JJG JJJG .

Vay I la iem tren ng thang qua A, song song vi BC sao cho IA 2BC=JJG JJJG 5) Ta co: FA GB AB FC AC 0 3FA 0 F A+ + = = JJJG JJJG JJJG JJJG JJJG G JJJG G 6) Goi G la trong tam tam giac ABC, ta co: 3KA GB GC 5GK 2GA 5GK+ + = =JJJG JJJG JJJG JJJG JJJG JJJG .

Vay K nam trong oan GA sao cho 2GK GA5

=JJJG JJJG 7) Goi M la trung iem AC, ta co: 3LA 2LB LC 0 + =JJJG JJJG JJJG G

3MA 2MB MC 2ML 2MA 2MB MA MB 2ML MA MB ML hay BA ML + = + + = = =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG . Vay L la iem th t cua hnh bnh hanh dng tren hai vect BA va BM

JJJG JJJG.

Bai 3: Cho tam giac ABC va iem D, E.

1) Chng minh neu OA OB OC 0+ + =JJJG JJJG JJJG G th O la trong tam tam giac ABC. 2) Xac nh M thoa:

a) MA 2MB 0+ =JJJG JJJG G . b) MA MB 2MC 0+ + =JJJG JJJG JJJG G . c) MA MB MD MD ME+ + = JJJG JJJG JJJG JJJG JJJG . d) 2MA 3MB MC 0+ =JJJG JJJG JJJG G .

3) Xac nh iem M thoa: a) MA 3MB 0 =JJJG JJJG G . b) MA MB MC AB AC+ + = +JJJG JJJG JJJG JJJG JJJG . c) 2MA 3MB 4MC 0 + =JJJG JJJG JJJG G .

4) Goi I la iem xac nh bi 5IA 7IB IC 0 =JJG JJG JJG G , G la trong tam tam giac ABC. a) Chng minh GI 2AB=JJG JJJG .


14

b) AI BG O= . Tnh OAOI

.

c) Xac nh iem thuoc ng thang d sao cho min

5MA 3MBJJJG JJJG . Hng dan:

1) Goi G la trong tam tam giac ABC.

( ) ( ) ( )OA OB OC 0 OG GA OG GB OG GC 0

3OG GA GB GC 0 3OG 0 hay O G

+ + = + + + + + = + + + = =

JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG JJJG GJJJG JJJG JJJG JJJG G JJJG G

2) a) 1MA 2MB 0 MB BA 2MB 0 3MB AB 0 MB AB3

+ = + + = = =JJJG JJJG G JJJG JJJG JJJG G JJJG JJJG G JJJG JJJG . b) ( )MA MB 2MC 0 2MI 2MC 0 I la tr ung iem AB MI MC 0+ + = + = + =JJJG JJJG JJJG G JJJG JJJG G JJJG JJJG G Vay M la trung iem IC. c) MA MB MC MD ME 3MG EM MD ED+ + = = + =JJJG JJJG JJJG JJJG JJJG JJJJG JJJG JJJG JJJG 1 1MG ED hay GM DE

3 3 = =JJJJG JJJG JJJJG JJJG ( ) G la tr ong tam ABC .

d) ( ) ( )2MA 3MB MC 0 2MA 3 MA AB MA AC 0 4MA 3AB AC 0+ = + + + = + =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG G JJJG JJJG JJJG G ( ) ( )1 1MA AC 3AB hay AM 3AB AC4 4 = = JJJG JJJG JJJG JJJJG JJJG JJJG .

3) a) ( )2MA 3MB 0 2MA 3 MA AB 0 AM 3AB = + = =JJJG JJJG G JJJG JJJG JJJG G JJJJG JJJG . b) ( )MA MB MC AB AC 3MG 2AI G la tr ong tam ABC, I la tr ung iem BC+ + = + = JJJG JJJG JJJG JJJG JJJG JJJJG JJG 2 2GM AI MG AI

3 3 = =JJJJG JJG JJJJG JJG .

c) ( ) ( ) ( )12MA 3MB 4MC 0 2MA 3 MA AB 4 MA AC 0 MA AB 4AC3 + = + + + = = JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG G JJJG JJJG JJJG 4) a) ( ) ( ) ( )5IA 7IB IC 0 5 GA GI 7 GB GI GC GI 0 = =JJG JJG JJG G JJJG JJG JJJG JJG JJJG JJG G

3GI 5GA 7GB GC 6GA 6GB v GA GB GC 0 = + + = + + + =JJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG G ( )GI 2 GB GA 2AB = =JJG JJJG JJJG JJJG b) OA AB 1GI 2AB GI // AB

OI GI 2= = =JJG JJJG .

c) Goi K la iem thoa: 5KA 3KB 0 =JJJG JJJG G ( ) ( )5MA 3MB 5 MK KA 3 MK KB 2MK 2MK = + + = =JJJG JJJG JJJJG JJJG JJJJG JJJG JJJJG JJJJG ( )

min5MA 3MB M la hnh chieu cua K len d JJJG JJJG .

Bai 4: Cho tam giac ABC, trong tam G.

1) Xac nh v tr iem M sao cho: a) MA MB 2MC 0+ + =JJJG JJJG JJJG G . b) MA MB MC 0 + =JJJG JJJG JJJG G .


15

c) MA 2MB 0+ =JJJG JJJG G . d) MA 2MB CB+ =JJJG JJJG JJJG .

2) Goi A la iem oi xng cua A qua B, B la iem oi xng cua B qua C va C la iem oi xng cua C qua A. Chng minh rang tam giac ABC va tam giac ABC co cung trong tam G. Hng dan:

1) a) Goi I la trung iem AB. Ta co: ( )MA MB 2MI MA MB 2MC 2 MI MC 0 MI MC 0+ = + + = + = + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG G JJJG JJJG G ( ) 1MC CI MC 0 CM CI2 + + = = JJJG JJG JJJG G JJJG JJG M la trung iem CI. b) MA MB MC 0 MA MB CM BA CM + = = =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG . Vay M xac nh bi he thc BA CM=JJJG JJJG . Do vay M la nh th t cua hnh bnh hanh ABCM. c) 1MA 2MB 0 MB BA 2MB 0 3BM BA hay BM BA

3+ = + + = = =JJJG JJJG G JJJG JJJG JJJG G JJJG JJJG JJJG JJJG .

Vay M la iem nam trong AB sao cho ABBM3

= . d) Ta co: ( )MA 2MB CB MB BA 2MB CB 3MB CB BA BA BC+ = + + = = = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG ( ) 23BM BA BC 2BJ J la tr ung iem AC BM BJ M la tr ong tam tam giac ABC

3 = + = = JJJG JJJG JJJG JJG JJJG JJG .

2) Ta co: ( )AB BC CA 0 2 AB BC CA 0+ + = + + =JJJG JJJG JJJG G JJJG JJJG JJJG G 2AB 2BC 2CA 0 AA BB CC 0 + + = + + =JJJG JJJG JJJG G JJJJG JJJG JJJG G Mat khac: AA BB CC 0 AG GA BG GB CG GC 0 + + = + + + + + =JJJJG JJJG JJJG G JJJG JJJJG JJJG JJJJG JJJG JJJJG G

( ) ( )GA GB GC GA GB GC 0GA GB GC 0 (1)

GA GB GC 0 (G la tr ong tam tam giac ABC)(1) ung khi G la tr ong tam tam giac A'B'

+ + + + = + + = + + =

JJJJG JJJJG JJJJG JJJG JJJG JJJG GJJJJG JJJJG JJJJG GJJJG JJJG JJJG G

C' .

Vay tam giac ABC va tam giac ABC co cung trong tam G. Bai 5:

1) Cho tam giac ABC eu canh a. Xac nh vect AB AC+JJJG JJJG va tnh moun cua vect nay. 2) Cho hnh vuong ABCD canh a. Xac nh vect ( )1 AB AC AD2 + +JJJG JJJG JJJG va tnh moun cua vect nay. 3) Gia s M va N theo th t la trung iem cua cac canh AD va BC cua t giac ABCD. Chng minh rang

( )1MN AB CD2

+ . Khi nao xay ra ang thc? Hng dan:

1) Ve trung tuyen AM, keo dai AM lay iem E sao cho: ME AM= . Ta co:

a 3AB AC 2AM AE AB AC 2 AM 2. a 3

2+ = = + = = =JJJG JJJG JJJJG JJJG JJJG JJJG JJJJG


16

2) ( ) ( )1 1 1AB AC AD AB AD AC AC AC AC2 2 2+ + = + + = + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG . Do o: ( )1 AB AC AD AC a 22 + + = =JJJG JJJG JJJG JJJG

3) Ta chng minh ( )1MN AB DC2= +JJJG JJJG JJJG Suy ra ( ) ( )1 1 1MN AB DC AB DC hay MN AB DC2 2 2= + + +JJJG JJJG JJJG JJJG JJJG ang thc xay ra khi AB // DC

JJJG JJJG hay ABCD la hnh thang ay AB, CD.

Bai 6: Cho t giac ABCD.

1) Tm iem co nh I va hang so k e he thc sau thoa vi moi M: a) MA MB 2MC kMI+ + =JJJG JJJG JJJG JJJG . b) 2MA 3MB MD kMI+ =JJJG JJJG JJJG JJJG . c) MA MB 2MC kMI =JJJG JJJG JJJG JJJG . d) MA 2MB 3MC 4MD kMI+ + =JJJG JJJG JJJG JJJG JJJG .

2) OA OB OC OD 0+ + + =JJJG JJJG JJJG JJJG G . Chng minh O xac nh duy nhat. 3) Neu ABCD la hnh bnh hanh. Vi moi M, hay tm K va iem I co nh thoa:

a) MA MB MC 3MD kMI+ + + =JJJG JJJG JJJG JJJG JJJG . b) MA 2MB kMI+ =JJJG JJJG JJJG . c) 2MA MB MC kMI+ =JJJG JJJG JJJG JJJG .

4) Xac nh v tr iem S e SA SB SC SD 0+ + + =JJG JJG JJG JJG G . 5) Goi G la trong tam t giac ABCD, A, B, C, D lan lt la trong tam cac tam giac BCD, ACD, ABD,

ABC. Chng minh rang G la iem chung cua cac oan thang AA, BB, CC, DD va cung la trong tam t giac ABCD. Hng dan:

1) a) I la trung iem AB. ( )MA MB 2MC kMI 2MI 2MC kMI 2 MI MC kMI+ + = + = + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG 4MO kMI =JJJG JJJG (O la trung iem IC). He thc cho cau a ung M I O, k 4 = b) Goi IS la iem thoa ( ) ( )2SA 3SB SD 0 2SA 3 SA AB SA AD 0+ = + + + =JJG JJG JJG G JJG JJG JJJG JJG JJJG G ( )1AS 3AB AD4 = JJJG JJJG JJJG khi S la iem co nh. Khi o ( ) ( ) ( )2MA 3MB MD kMI 2 MS SA 3 MS SB MS SD kMI+ = + + + + =JJJG JJJG JJJG JJJG JJJG JJG JJJG JJG JJJG JJG JJJG 4MS 2SA 3SB SD kMI 4MS kMI + + = =JJJG JJG JJG JJG JJJG JJJG JJJG . He thc cho cau b ung M khi I S,k 4 = .

c) Goi J la iem thoa 1JA JB 2JC 0 JC BA2

= =JJG JJG JJG G JJG JJJG (J xac nh)


17

( ) ( ) ( )MA MB 2MC kMI MJ JA MJ JB 2 MJ JC kMI = + + + =JJJG JJJG JJJG JJJG JJJG JJG JJJG JJG JJJG JJG JJJG 2MJ JA JB 2JC kMI 2MJ kMI + = =JJJG JJG JJG JJG JJJG JJJG JJJG He thc cho cau c ung M khi I J ,k 2 = .

2) Goi N, L la trung iem AB, CD. OA OB OC OD 0 2ON 2OL 0 ON OL 0+ + + = + = + = JJJG JJJG JJJG JJJG G JJJG JJJG G JJJG JJJG G O la trung iem NL O xac nh duy nhat

3) a) Goi G la trong tam tam giac ABC, ta co: MA MB MC 3MG+ + =JJJG JJJG JJJG JJJJG ( ) ( )MA MB MC 3MD 3 MG MD 3 2MI 6MI + + + = + = =JJJG JJJG JJJG JJJG JJJJG JJJG JJJG JJJG (I la trung iem GD). Vay k 6= . b) Goi I la iem thoa 2IA 2IB 0 AI AB

3+ = =JJG JJG G JJG JJJG .

( )MA 2MB 1 2 MI 3MI+ = + =JJJG JJJG JJJG JJJG . Vay k 3= . c) I la iem nh bi 12IA IB IC 0 2IA CB 0 hay AI CB

2+ = + = =JJG JJG JJG G JJG JJJG G JJG JJJG .

( )2MA MB MC 2 1 1 MI 2MI+ = + =JJJG JJJG JJJG JJJG JJJG . Vay k 2= .

4) Goi P, Q lan lt la trung iem BD, AC. Ta co:

( )SA SC 2SN SA SB SC SD 2 SN SMSB SD 2SM

+ = + + + = ++ =

JJG JJG JJG JJG JJG JJG JJG JJG JJJGJJG JJG JJJG (xem cac bai tap trc) SM SN 0 + =JJJG JJG G

S la tr ung iem MN 5) V G la trong tam t giac ABCD nen GA GB GC GD 0+ + + =JJJG JJJG JJJG JJJG G (1)

Mat khac A la trong tam tam giac BCD nen ( ) ( ) ( )GB GC GD GA A B GA A C GA A D GB GC GD 3GA + + = + + + + + + + =JJJG JJJG JJJG JJJJG JJJG JJJJG JJJJG JJJJG JJJJG JJJG JJJG JJJG JJJJG (2) T (1) va (2) suy ra GA 3GA 0 GA 3GA + = = JJJG JJJJG G JJJG JJJJG . Vay 3 iem G, A, A thang hang. Chng minh tng t ta co: G, B, B thang hang; G, C, C thang hang. Hay AA, BB, CC ong quy tai G. T (1) va (2) suy ra ( )GA GB GC GD 3 GA GB GC GD 0 + + + = + + + =JJJG JJJG JJJG JJJG JJJJG JJJJG JJJJG JJJJG G Hay GA GB GC GD 0 G la tr ong tam t giac A B C D + + + = JJJJG JJJJG JJJJG JJJJG G . Bai 12: Cho tam giac ABC, k la hang so, M la iem di dong sao cho v MN 2MA 3MB kMC= = +G JJJG JJJG JJJG JJJG . 1) Vi k 1= . Chng minh MNJJJG co phng khong oi. 2) Vi k 1 . Chng minh MN luon i qua mot iem co nh.

Hng dan: 1) ( ) ( )k 1 : v 2MA 3MB MC 2MA 3 MA AB MA AC 3AB AC= = + = + + + = +G JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG vect khong oi. Vay MN

JJJG co phng khong oi.


18

2) Goi I: ( ) ( ) ( )2IA 3IB kIC 0 2 CA CI 3 CB CI kCI 0 1 k CI 3CB 2CA + = = = JJG JJG JJG G JJJG JJG JJJG JJG JJG G JJG JJJG JJJG 3CB 2CACI I co nh, k 1

1 k = .

Va: ( ) ( ) ( ) ( ) ( )MN 2 MI IA 3 MI IB k MI IC k 1 MI 2IA 3IB kIC k 1 MI= + + + + = + + = JJJG JJJG JJG JJJG JJG JJJG JJG JJJG JJG JJG JJG JJJG . Vay ( )MN k 1 MI MN // MI MN i qua I co nh= JJJG JJJG .

Bai 13: Cho tam giac ABC, trong tam G. Cac iem M, N thoa man: 13MA 4MB 0,CN BC2

+ = =JJJG JJJG G JJJG JJJG . Chng minh MN i qua trong tam G cua tam giac ABC.

Hng dan: 33MA 4MB 0 MB AB7

1 3CN BC BN BC2 2

+ = =

= =

JJJG JJJG G JJJG JJJG

JJJG JJJG JJJG JJJG

Ta co: 3 3MN MB BN AB BC7 2

= + = +JJJG JJJG JJJG JJJG JJJG (1) G la trong tam tam giac ABC nen :

( )3 2 3 2 1 2 1MG MB BG AB BE AB . BA BC AB BC7 3 7 3 2 21 3= + = + = + + = +JJJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG (2) T (1) va (2) suy ra 9MN MG

2=JJJG JJJJG .

Vay ba iem M, N, G thang hang. Bai 14: Cho tam giac ABC. 1) MN v 2MA 3MB kMC= = + +JJJG G JJJG JJJG JJJG .

a. Khi k 5 . Chng minh gia cua MNJJJG luon qua mot iem co nh. b. Tm k e MN

JJJG la mot vect khong oi.

2) Lay E, F tren tam giac ABC sao cho ( )1 1AE AB, AF AC k 0; 1k k 1

= = +JJJG JJJG JJJG JJJG

. Chng minh EF luon qua

mot iem co nh. Hng dan:

1) a) Goi I la iem thoa ( ) ( )2IA 3IB kIC 0 2IA 3 IA AB k IA AC 0+ + = + + + + =JJG JJG JJG G JJG JJG JJJG JJG JJJG G ( ) ( )3AB kAC5 k IA 3AB kAC IA k 5

5 k + = = +JJJG JJJGJJG JJJG JJJG JJG

I co nh.

( ) ( ) ( ) ( )MN v 2 MI IA 3 MI IB k MI IC 5 k MI MN // MI= = + + + + + = + JJJG G JJJG JJG JJJG JJG JJJG JJG JJJG JJJG JJJG hay MN qua I co nh. b) ( ) ( ) ( )MN 2MA 3 MA AB k MA AC 5 k MA 3AB kAC= + + + + = + + +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG MN

JJJG khong oi khi 5 k 0 k 5+ = = .

2) ( ) ( )AC k 1 AF kAF AF k AE EF AF AC AB kEF AF= + = + = + + = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG


19

Hay kEF AC AB AF BC AF= = JJJG JJJG JJJG JJJG JJJG JJJG Ve AI BC=JJG JJJG th AI AF kEF kEF FI = = JJG JJJG JJJG JJJG JJG E, F, I thang hang EF qua I co nh. Bai 15: Cho hnh bnh hanh ABCD. 1) Goi I, F, K la cac iem xac nh bi AI AB, AF AC, AK AD= = = JJG JJJG JJJG JJJG JJJG JJJG . Chng minh ieu kien can va u e I, F, K thang hang la ( )1 1 1 , , 0= + . 2) Goi M, N la 2 iem lan lt tren oan AB va CD sao cho AM 1 CN 1,

AB 3 CD 2= = . Goi G la trong tam tam giac

BMN. a) Tnh AN, AG theo AB, AC

JJJG JJJG JJJG JJJG.

b) Goi H la iem nh bi BH kBC=JJJG JJJG . Tnh AH theo AB, AC va kJJJG JJJG JJJG . Tm k e AH qua G. Hng dan:

1) Ta co: KI AI AK AB AD= = JJG JJG JJJG JJJG JJJG KF AF AK AC AD= = JJJG JJJG JJJG JJJG JJJG Ma AC AB AD= +JJJG JJJG JJJG Vay ( )KF AB AD= + JJJG JJJG JJJG ieu kien can va u e K, I, F thang hang va ton tai so k sao cho KF kKI=JJJG JJG ( ) ( ) ( )AB AD k AB k AD k AB k AD 0 + = + + =JJJG JJJG JJJG JJJG JJJG JJJG G (1) Do AB, AD

JJJG JJJG khong cung phng nen ( )1 k 0 k k 1 = = + = = =

( )1 1 1 , , 0 = + (pcm). 2) a) Ta co:

AD AC 2AN 1AC AB 2AN AC AN AC AB

2AD AB AC

+ = = = + =

JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJGJJJG JJJG JJJG

Va ta co: 13AG AB AM AN AB AN AB3

= + + = + +JJJG JJJG JJJJG JJJG JJJG JJJG JJJG

4 1 53AG AB AC AB AB AC3 2 6

5 1AG AB AC18 3

= + = +

= +


JJJG JJJG JJJG

b) ( )AH AB BH AB kBC AB kBA kAC 1 k AB kAC= + = + = + + = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG ieu kien can va u e AG qua H la AGJJJG cung phng AHJJJG ( )5 1 6k 1 k k

18 3 11 = = .

Bai 16: Cho tam giac ABC. 1) Goi I la trung iem cua BC, D va E la hai iem sao cho BD DE EC= =JJJG JJJG JJJG .


20

a) Chng minh rang AB AC AD AE+ = +JJJG JJJG JJJG JJJG . b) Tnh AS AB AD AC AE theo AI= + + +JJJG JJJG JJJG JJJG JJJG JJG , suy ra A, I , S thang hang.

2) Goi M la iem nh bi BM BC 2AB= JJJG JJJG JJJG , N la iem nh bi CN xAC BC= JJJG JJJG JJJG . a) Xac nh x e A, M, N thang hang.

b) Xac nh x e MN qua iem gia I cua BC. Tnh IMIN

.

Hng dan: 1) a) Ta co:

( )( )

AB AC 2AI I la tr ung iem BCAB AC AD AE

AD AE 2AI I la tr ung iem DE

+ = + = ++ =

JJJG JJJG JJGJJJG JJJG JJJG JJJG

JJJG JJJG JJG

b) ( ) ( )AS AB AC AD AE 2AI 2AI 4AI= + + + = + =JJJG JJJG JJJG JJJG JJJG JJG JJG JJG V AS 4AI=JJJG JJG nen AS va AIJJJG JJG cung phng hay A, S, I thang hang. 2) a) Ta co: BM BC 2AB 2AB BC BM= = JJJG JJJG JJJG JJJG JJJG JJJG ( ) ( )Hay 2AB BA AC BA AM AC AM AM AC 2AB= + + = = JJJG JJJG JJJG JJJG JJJJG JJJG JJJJG JJJJG JJJG JJJG (1) ( ) ( )CN xAC BC xAC CN CB CA AN CA AB= = = + +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG Hay xAC AN AB AN xAC AB= = +JJJG JJJG JJJG JJJG JJJG JJJG (2) e A, M, N thang hang AM, AN JJJJG JJJG cung phng AM kAN =JJJJG JJJG Chon ( ) ( )k 2 th AM 2AN AM 2AN 0 AC 2AB 2 xAC AB 0= = + = + + =JJJJG JJJG JJJJG JJJG G JJJG JJJG JJJG JJJG G ( ) 12x 1 AC 0 2x 1 0 x

2 + = + = = JJJG G

Vay vi 1x2

= th A, M, N thang hang. b) ( ) ( ) ( )BM BC 2AB 2 BI IA BI IC BI IM 0= + + + + =JJJG JJJG JJJG JJG JJG JJG JJG JJG JJJG G 2IB 2IA IC IM IM 2IA 3IC = + = +JJG JJG JJG JJJG JJJG JJG JJG ( )CN xAC BC x CI IA CI IN CI IB 0= + + + =JJJG JJJG JJJG JJG JJG JJG JJG JJG JJG G ( )xCI xIA IN IB IN xIA x 1 IC = + = + JJG JJG JJG JJG JJG JJG JJG e M, N, I thang hang IM va IN cung phng IM kIN =JJJG JJG JJJG JJG . Chon ( )k 5 th IM 5IN IM 5IN 0 2IA 3IC 5 xIA x 1 IC 0 = = + = + + =

JJJG JJG JJJG JJG G JJG JJG JJG JJG G

( ) 22 x 1 3x 0 x5

+ = =

Vay vi 2x5

= th M, N, I thang hang.

IM 2IA 3IC2 IMKhi x th IM 5IN 52 35 ININ IA IC

5 5

= += = = =

JJJG JJG JJGJJJG JJG

JJG JJG JJG


21

Bai 17: Cho ba iem co nh A, B, C va ba so thc sao cho a b c 0+ + . 1) Chng minh rang ch co mot iem G thoa man he thc aGA bGB cGC 0+ + =JJJG JJJG JJJG G . 2) Goi M va P la hai iem di ong sao cho MP aMA bMB cMC= + +JJJG JJJG JJJG JJJG . Chng minh rang ba iem G, M, P thang hang.

Hng dan: 1) Ta co: ( ) ( )aGA bGB cGC 0 aGA b GA AB c GA AC 0+ + = + + + + =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG G ( ) bAB cACa b c GA bAB cAC AG

a b c+ + + = = + +

JJJG JJJGJJJG JJJG JJJG JJJG khong oi.

Do o G c xac nh va duy nhat. 2) ( ) ( ) ( )MP aMA bMB cMC aMP a MG GA b MG GB c MG GC= + + = = + + + + +JJJG JJJG JJJG JJJG JJJG JJJJG JJJG JJJJG JJJG JJJJG JJJG ( ) ( ) ( )MP a b c MG aGA bGB cGC a b c MG= + + + + + = + +JJJG JJJJG JJJG JJJG JJJG JJJJG . Vay M, P ,G thang hang. Bai 18: Cho tam giac ABC. 1) Goi M la trung iem BC, I va J la hai iem xac nh bi AI AB va AJ AC= = JJG JJJG JJG JJJG . Tm he thc lien he gia va e AM cat IJ tai trung iem cua AM. 2) Goi P la iem lu ong. Dng vect PQ 2PA 3PB PC= + JJJG JJJG JJJG JJJG . Chng minh PQ i qua mot iem co nh khi P thay oi . Goi H la trung iem cua CQ. Chng minh rang PH i qua mot iem co nh khi P thay oi.

Hng dan: a. Neu 0 0 = = : khong thoa man gia thiet. Do vay 0 va 0 .

Ta co: ( )1 1 1 1 1 1AM AB AC AI AI AI AI2 2 2 2 = + = + = + JJJJG JJJG JJJG JJG JJG JJG JJG

Do o AM se cat IJ tai K ma 1 1AM AK2 2

= + JJJJG JJJG

(tam t c)

Vay K la trung iem cua AM th 1 1 1 1

2 42 2

+ = + = b. Goi E la iem nh bi ( ) ( )2EA 3EB EC 0 2EA 3 EA AB EA AC 0+ = + + + =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG G

5AE 3AB AC E co nh = JJJG JJJG JJJG . Ta chng minh PQ qua E. ( ) ( ) ( )PQ 2PA 3PB PC 2 PE EA 3 PE EB PE EC 4PE 2EA 3EB EC 4PE= + = + + + + = + + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG P, Q, E thang hang PQ qua iem co nh E. Ta co: ( ) ( ) ( )1 1 1 5PH PQ PC 2PA 3PB 2 3 PJ PJ2 2 2 2= + = + = + =JJJG JJJG JJJG JJJG JJJG JJG JJG P, H, J thang hang PH qua iem co nh J. Bai 19: Cho bon iem A, B, C, M thoa man MA 2MB 3MC 0+ =JJJG JJJG JJJG G . Chng minh A, B, C thang hang.


22

Hng dan:

( ) ( ) 3MA 2MB 3MC 0 MA 2 MA AB 3 MA AC 0 2AB 3AC 0 AB AC2+ = + + + = = =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG G JJJG JJJG G JJJG JJJG A, B, C thang hang. Bai 20: Cho tam giac ABC, M va N thay oi sao cho MN 2MA 3MB MC= + JJJG JJJG JJJG JJJG 1) Tm iem I thoa man 2IA 3IB IC 0+ =JJG JJG JJG G . 2) Chng minh ng thang MN luon i qua mot iem co nh.

Hng dan: 1) ( ) ( )0 2IA 3IB IC 2IA 3 IA AB IA AC 4IA 3AB AC= + = + + + = + G JJG JJG JJG JJG JJG JJJG JJG JJJG JJG JJJG JJJG ( )1IA AC 3AB4 = JJG JJJG JJJG : ang thc xac nh iem I 2) ( ) ( ) ( ) ( )MN 2MA 3MB MC 2 MI IA 3 MI IB MI IC 4MI 2IA 3IB IC 4MI= + = + + + + = + + =JJJG JJJG JJJG JJJG JJJG JJG JJJG JJG JJJG JJG JJJG JJG JJG JJG JJJG M, N, I thang hang hay MN i qua iem I co nh. Bai 21: Cho tam giac ABC, goi I, J la hai iem xac nh bi IA 2IB; 3JA 2JC 0= + =JJG JJG JJG JJG G 1) Tnh IJ theo AB va AC

JJG JJJG JJJG.

2) Chng minh ng thang IJ i qua trong tam G cua tam giac ABC. Hng dan:

1) IJ AJ AI= JJG JJG JJG ( )AI 2BI 2 BA AI AI 2AB= = + =JJG JJG JJJG JJG JJG JJJG ( ) 23JA 2JC 0 3JA 2 JA AC 0 AJ AC5+ = + + = =JJG JJG G JJG JJG JJJG G JJG JJJG Vay 2IJ AC 2AB

5= JJG JJJG JJJG

2) Ta co: ( )1AG AB AC3= +JJJG JJJG JJJG Mat khac: ( )1IG AG AI AC 5AB3= = +JJG JJJG JJG JJJG JJJG 6IJ IG

5 = I, J, G thang hang.

Bai 22: Cho tam giac ABC, hai iem M, N thay oi sao cho MN 2MA MB MC= +JJJG JJJG JJJG JJJG 1) Dng iem I sao cho 2IA IB 3IC 0 + =JJG JJG JJG G . 2) Chng minh rang MN i qua mot iem co nh khi M thay oi. 3) Goi P la trung iem cua BN. Chng minh MP i qua mot iem co nh khi M thay oi.

Hng dan: 1) ( ) ( ) ( )2IA IB 3IC 0 2 OA OI OB OI 3 OC OI 0 2OA OB 3OC 4OI + = + = + =JJG JJG JJG G JJJG JJG JJJG JJG JJJG JJG G JJJG JJJG JJJG JJG Neu chon O C th 4CI 2CA CB = JJG JJJG JJJG : ang thc nay hoan toan xac nh I.


23

2) ( ) ( ) ( )MN 2MA MB 3MC 2 MI IA MI IB 3 MI IC MN 4MI= + = + + + + =JJJG JJJG JJJG JJJG JJJG JJG JJJG JJG JJJG JJG JJJG JJJG M, N, I thang hang hay MN qua I co nh. 3) ( ) ( )1 1MP MB MN 2MA 3MC2 2= + = +JJJG JJJG JJJG JJJG JJJG Goi J la iem thoa man 2JA 3JC 0+ =JJG JJG G , J xac nh. ( ) ( )1 5MP 2 MJ JA 3 MJ JC MJ2 2 = + + + = JJJG JJJG JJG JJJG JJG JJJG . Vay M, P, J thang hang hay MP qua J co nh. Bai tap: Bai 1: Cho tam giac ABC va tam giac ABC. Goi G, G lan lt la trong tam cua hai tam giac tren.

1. Chng to AA BB CC 3GG + + =JJJJG JJJG JJJG JJJJG suy ra ieu kien can va u e hai tam giac co cung trong tam. 2. Gia s A, B, C lan lt thuoc canh BC, CA, AB. Chng minh rang ieu kien can va u e hai tam

giac ABC va ABC co cung trong tam la BA CB ACBC CA AB

= = . Bai 2: Cho tam giac ABC, ben ngoai tam giac ve cac hnh bnh hanh ABIF, BCPQ, CARS. Chng minh rang RF IQ PS 0+ + =JJJG JJG JJG G .

Hng dan: RF FI IQ QP PS SR 0+ + + + + =JJJG JJG JJG JJJG JJG JJG G RF AB IQ BC PS CA 0 + + + + + =JJJG JJJG JJG JJJG JJG JJJG G ( )RF IQ PS AB BC CA 0 + + + + + =JJJG JJG JJG JJJG JJJG JJJG G RF IQ PS 0 + + =JJJG JJG JJG G Bai 3: Cho hnh thang ABCD, AC cat BD tai O. Qua O ve ng thang MN song song hai ay AD, BC. at

a AB , b CD= =JJJG JJJG . Chng minh bAB aDCMNa b+= +

JJJG JJJGJJJG.

Hng dan:

MA NB OA OB AB aMN // AB // CDMB NC OC OC DC b

= = = = = aOA ODbOMa aMA MD 1

b b MN ON OMa aOB OCNB NC

bb ON a1b

+ = = + = += = +

JJJG JJJGJJJG

JJJG JJJGJJJG JJJG JJJG

JJJG JJJGJJJG JJJGJJJG

(pcm)

Bai 4: Cho tam giac ABC. M va N la cac iem xac nh bi 3MA 4MB NB 3NC 0+ = =JJJG JJJG JJJG JJJG G , AC cat MN tai P. Tnh t so PA

PC.

Hng dan: Gia s AC kAP=JJJG JJJG .


24

Ta co: ( ) 73MA 4MB 0 3AM 4 AB AM 0 AB AM4+ = + = =JJJG JJJG G JJJJG JJJG JJJJG G JJJG JJJJG ( )1 1 7 3k 7 3kNB 3NC AN AB 3AC . .AM AP AN AM AP2 2 4 2 8 2= = = + = +JJJG JJJG JJJG JJJG JJJG JJJJG JJJG JJJG JJJJG JJJG V M, N, P thang hang nen 3k 7 51 k

2 8 4 = = .

Vay PA 4PC

= .

Bai 5: Cho hnh bnh hanh ABCD. M va N la cac iem thoa man 1 1AM AB; DN DC3 2

= =JJJJG JJJG JJJG JJJG , G la trong tam

tam giac MNB, AG cat BC tai I. Tnh cac t so AG BI,GI IC

.

Hng dan:

at AG BI,GI IC

= = . Ta co:

( )AI 1 1 1 1 1 1 11AI .AG . AM AN AB AB AD DC AB AB ADAG 3 3 3 2 3 3 + + + = = + + = + + + = + JJG JJJG JJJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG ( )11 1 1

AI AB AD18 3 + + = +

JJG JJJG JJJG (1)

AB ACIB IC AI AD AB

1 1+ = = = ++ +

JJJG JJJGJJG JJG JJG JJJG JJJG (2)

T (1) va (2) suy ra

( )11 1 AG 1111118 GI 77

61 BI 653 1 IC 5

+ = = = + == = +

Bai 6: Cho tam giac ABC. 1. Goi I la iem tren canh BC sao cho BI 2IC=JJG JJG . Hay tnh vect AI theo hai vect AB va ACJJG JJJG JJJG . 2. Goi N, P lan lt la trung iem cua CA, AB. at BN a,CP b= =JJJG G JJJG G . Tnh AB, BC,CA theo a va bJJJG JJJG JJJG G G .

Hng dan: 1. Ta co:

( )2

AI AB BI AB BC13 AI AB 2AC

1 3AI AC CI AB BC3

= + = + = + = + =

JJG JJJG JJG JJJG JJJGJJG JJJG JJJG

JJG JJJG JJG JJJG JJJG

2. Ta co:

ABAB AC CP PB AC b 4 2 2 4 2 22 AB a b; CA a b; BC a b3 3 3 3 3 3CACA CN NB BA a AB

2

= + + = + + = = + = = + + =

JJJGJJJG JJJG JJJG JJJG JJJG GJJJG G G JJJG G G JJJG G G

JJJGJJJG JJJG JJJG JJJG G JJJG


25

Bai 7: Cho bon iem A, B, C, D. Ta dng cac iem M , N, P thoa man AM 2AB, AN 2AC,= =JJJJG JJJG JJJG JJJG AP 2AD=JJJG JJJG . 1. Tnh MN theo BC, NP theo CD


2. Chng minh rang M ,N, P thang hang neu va ch neu B, C, D thang hang. Hng dan:

1. Ta co:

AM 2AB MM 2BC

AN 2AC NP 2CD

= = = =

JJJJG JJJG JJJJG JJJGJJJG JJJG JJJG JJJG

M, N, D thang hang MN kNP BC kCD pcm = = JJJG JJJG JJJG JJJG

TAM TY C Bai 1: Cho ba iem A, B, C ng vi ba so , , . 3) Neu 0 + + th ton tai duy nhat iem I thoa: IA IB IC 0 + + =JJG JJG JJG G khi

( )v MA MB MC MI, M= + + = + + G JJJG JJJG JJJG JJJG . 4) Neu 0 + + = th MA MB MC + + JJJG JJJG JJJG la mot vect khong oi. 5) nh M e

min2MA 3MB MC+ JJJG JJJG JJJG trong tam giac ABC.

6) Chng minh rang ton tai duy nhat iem I thoa IA IB 0 + =JJG JJG G . Suy ra rang vi M bat ky, ta co: ( )MA MB MI + = + JJJG JJJG JJJG .

Hng dan: 1) ( ) ( ) ( )IA IB IC 0 AI AB AI AC AI 0 AI AB AC + + = + + = + + = + JJG JJG JJG G JJG JJJG JJG JJJG JJG G JJG JJJG JJJG ( ) ( )1AI AB AC 0 = + + + + + JJG JJJG JJJG I xac nh duy nhat. Ta co: ( ) ( ) ( )v MA MB MC MA MA AB MA AC MA AB AC= + + = + + + + = + + + + G JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG v AB AC, M= + G JJJG JJJG Mat khac: ( ) ( ) ( )v MI IA MI IB MI IC= + + + + +G JJJG JJG JJJG JJG JJJG JJG

( )( )

MI IA IB IC

MI

= + + + + + = + +

JJJG JJG JJG JJGJJJG

Bai toan tren I goi la tam t c cua he iem A, B, C. 2) at ( ) ( ) ( )minf M MA MB MC MI 0 f M khi M I= + + = + + JJJG JJJG JJJG . Do o f(M) la mot vect khong oi. 3) Goi I la iem thoa 2MA 3MB MC+ JJJG JJJG JJJG . I ton tai v 2 3 1 4 0 + + = + = . Theo cau 2 ( ) ( ) ( )minf M MI 4MI 0 f M khi M I= + + = JJJG JJJG


26

4) ( ) ( )IA IB 0 IA IA AB 0 IA AB 0 AI AB + = + + = + + = = +JJG JJG G JJG JJG JJJG G JJG JJJG G JJG JJJG . Do A, B co nh, vect AB

+

JJJG khong oi nen ton tai mot iem I duy nhat thoa IA IB 0 + =JJG JJG G .

Ta cung co: ( ) ( ) ( )MA MB MI IA MI IB MI IA IB + = + + + = + + + JJJG JJJG JJJG JJG JJJG JJG JJJG JJG JJG . Vay ( )MA MB MI + = + JJJG JJJG JJJG . Bai 2: Cho tam giac ABC. 3) Goi I la iem xac nh bi ( )IA IB IC 0 0; 0; 0; 0 + + = + + + + + JJG JJG JJG G . Goi D, E, F lan lt la giao iem cua IA, IB, IC vi BC, CA, AB. Chng minh rang

DB DC EC EA FB FA 0 + = + = + =JJJG JJJG JJJG JJJG JJG JJJG G (1). 4) ao lai D, E, F la cac iem thoa (1) . Chng minh AD, BE, CF ong quy tai I xac nh nh cau 1. 5) Ap dung: Goi M la iem tuy y trong tam giac ABC eu; x, y, z lan lt la khoang cach t M ti BC, CA, AB. Chng minh rang xMA yMB zMC 0+ + =JJJG JJJG JJJG G . Hng dan: 3) Goi D la iem thoa ( ) ( ) ( )DB DC 0 IB IC ID tam t c IA ID 0 + = + = + + + =JJJG JJJG G JJG JJG JJG JJG JJG G I, A, D thang hang D chnh la giao iem cua IA vi BC. Tng t cho E, F. 4) Xet I la tam t c cua ( ) ( ) ( )A , B ,C .

( )( ) ( )

( ) ( ) ( )

IA IB IC 0 IA ID 0

IB IE 0; IC IF 0

I, D, A ; I, E, B ; I, F,C la nhng bo ba iem thang hang.

+ + = + + = + + = + + =

JJG JJG JJG G JJG JJG GJJG JJG G JJG JJG G

Do o: DA, EB, FC ong quy tai I. 5) Goi D AM BC; E BM CA; F CM AB= = = Ta co: ABD ABM

ACD ACM

dt dtDB z yDB zDC yDB zDC 0DC dt dt y

= = = = + =JJJG JJJG G (1) Tng t ta co: zEC xEA 0+ =JJJG JJJG (2) xFA yFB 0+ =JJJG JJG G (3) T (1), (2) va (3) suy ra AD, BE, CF ong quy tai M thoa man xMA yMB zMC 0+ + =JJJG JJJG JJJG G . Bai 3: Cho tam giac ABC, ng tron noi tiep (co tam I), tiep xuc vi BC, CA, AB lan lt tai D, E, F. AA, BB, CC la cac ng phan giac trong cua tam giac ABC ( )A BC; B AC; C AB va a, b, c la o dai cua BC, CA, AB. 3) Chng minh aIA bIB cIC 0+ + =JJG JJG JJG G . 4) Chng minh AD, BE, CF ong quy tai J thoa 1 1 1JA JB JC 0

p a p b p c+ + =

JJG JJG JJG G.


27

5) Chng minh tam giac ABC eu neu xay ra mot trong hai he thc sau ay: AA BB CC 0 + + =JJJJG JJJG JJJG G ; AD BE CF 0+ + =JJJG JJJG JJJG G . Hng dan:

3) AA, BB, CC ong quy tai I va A B AB c bA B cA C 0A C AC b = = + =

JJJG JJJJG G.

9 Cach 1: Tng t ta co cB C aB A 0 + =JJJG JJJG G va aC A bC B 0 + =JJJG JJJG G Vay aIA bIB cIC 0+ + =JJG JJG JJG G . 9 Cach 2: Neu I la iem ben trong tam giac ABC bat ky, ta luon co I la tam t c cua ( )IBCA S , ( ) ( )ICA IABB S ,C S . Ngha la IBC ICA IAB 1 1 1S .IA S .IB S .IC 0 r a.IA r b.IB r c.IC 02 2 2+ + = + + =

JJG JJG JJG G JJG JJG JJG G

aIA bIB cIC 0+ + =JJG JJG JJG G . 4) Ta co: BD BF p b; CD CE p c; AE AF p a= = = = = = ( ) ( ) 1 1p c DB p b DC 0 DB DC 0

p b p c + = + =

JJJG JJJG G JJJG JJJG G

Tng t ta co: 1 1 1 1

EC EA 0 va FA FB 0p c p a p a p b

+ = + = JJJG JJJG G JJJG JJG G

.

Do o DA, EB, FC ong quy tai J thoa 1 1 1JA JB JC 0p a p b p c

+ + = JJG JJG JJG G

5) Chng minh bo e: iem M nam trong tam giac ABC, AM cat BC tai D, BM cat CA tai E, CM cat AB tai F. Neu ta co AD BE CF 0 th M G + + = JJJJG JJJG JJJG G (G la trong tam tam giac ABC). Bo e nay ang tren tap ch toan hoc tuoi tre. at ABC 1 MBC 2 MCA 3 MABS dt ,S dt ,S dt ,S dt = = = = Ta co: 1 2 3S .MA S .MB S .MC 0+ + =

JJJG JJJG JJJG G (tam t c cach 2 cau 1)

Mat khac: AD BE CFAD BE CF 0 .MA .MB .MC 0AM BM CM

+ + = + + =JJJJG JJJG JJJG G JJJG JJJG JJJG G

1 2 3

S S S.MA .MB .MC 0

S S S S S S + + =

JJJG JJJG JJJG G.

1 2 3

1 1 1.MA .MB .MC 0S S S S S S

+ + = JJJG JJJG JJJG G

.

Do tnh chat cac giao iem AM BC, BM CA,CM AB nen ta co:

( ) ( ) ( )( ) ( ) ( )

31 21 1 2 2 3 3

1 2 3

1 2 3 2 1 3 3 2 1 1 2 2 3 3 1 1 2 3

11 1S SS S S S

S S S S S S S S SS S S

S S S S S S S S S S S S S S S S S S

MA MB MC 0 M G. Bo e c chng minh.

= = = = + = + = + = = = = + + = JJJG JJJG JJJG G

Vay neu ta co AA BB CC 0 tam ng tr on noi t iep cung la tr ong tam cua tam giac ABC + + = JJJJG JJJG JJJG G tam giac ABC eu .


28

Vay neu ta co AD BE CF 0 AD, BE,CF la cac tr ung tuyen cua tam giac ABC+ + = JJJG JJJG JJJG G DB DC = ; EC EA; FA FB= = p b p c; p c p a; p a p b a b c tam giac ABC eu = = = = = . Bai 4: Cho tam giac ABC, hay dng I, J, K, L thoa: 3) IA 3IB AC =JJG JJG JJJG . 4) JA JB 2JC 0 + =JJG JJG JJG G . 5) KA 2KB 2CB+ =JJJG JJJG JJJG . 6) LA 3LB LC 0+ =JJJG JJJG JJJG G . 7) IA IB 2IC AB + =JJG JJG JJG JJJG . 8) JA JB JC AB 2AC+ + = JJG JJG JJG JJJG JJJG . 9) KA KB 2KC 0+ + =JJJG JJJG JJJG G . 10) 3LA 2LB LC 0 + =JJJG JJJG JJJG G . Bai 5: Cho tam giac ABC, tm tap hp nhng iem M sao cho:

1) 3MA MB MC MB MC2

+ + = +JJJG JJJG JJJG JJJG JJJG . 2) MA BC MA MB+ = JJJG JJJG JJJG JJJG . 3) 2MA MB 4MB MC+ = JJJG JJJG JJJG JJJG . 4) 4MA MB MC 2MA MB MC+ + = JJJG JJJG JJJG JJJG JJJG JJJG . Bai 6: Cho t giac ABCD, hay tm so k va iem I co nh sao cho tong cac vect sau co the viet dang kMI, MJJJG . 2) MA MB MC kMI+ =JJJG JJJG JJJG JJJG . 3) MA MB 2MC kMI+ + =JJJG JJJG JJJG JJJG . 4) MA MB MC MD kMI+ + + =JJJG JJJG JJJG JJJG JJJG . 5) 2MA 3MC 2MD kMI + =JJJG JJJG JJJG JJJG . Bai 7: Cho hnh vuong ABCD canh a, M la iem bat ky. Chng minh rang cac vect sau day khong oi va moun cua chung la: 3) 2MA MB MC 2MD 3a+ =JJJG JJJG JJJG JJJG . 4) 3MA MB 2MC a 13 =JJJG JJJG JJJG . 5) 4MA MB MC MD 2a 2 =JJJG JJJG JJJG JJJG .

VAN E 4: CHNG MINH IEM THANG HANG, NG THANG QUA IEM CO NH, NG SONG SONG

Bai 1: Cho tam giac ABC, G la trong tam.


29

3) I va J la hai iem thoa IC IB IA 0 va JA JB 3JC 0 + = + =JJG JJG JJG G JJG JJG JJG G . Chng minh I, G, B thang hang va IJ cung phng AC . 4) Goi M la iem di ong tren BC, ve MP, MQ lan lt song song AC, AB cat AB, AC theo th t tai P, Q. ve hnh bnh hanh BMPR va CMQS. Tm tap hp trung iem I cua RS. 5) Goi N la iem xac nh bi 3NA NB 2NC 0 + =JJJG JJJG JJJG G . Tm AN BC, NB CA, NC AB . 6) Neu AB 3, AC 4= = , phan giac trong AD cua goc BAC cat trung tuyen BK tai L. Tnh AD

AL.

Hng dan: 3) ( ) ( ) ( ) ( )IC IB IA 0 IG GC IG GB IG GA 0 IG GC GA GB 0 + = + + + + = + + =JJG JJG JJG G JJG JJJG JJG JJJG JJG JJJG G JJG JJJG JJJG JJJG G

( )IG 2GB v GA GC GB = + = JJG JJJG JJJG JJJG JJJG I, G, B thang hang. IC IB IA 0 BC IA 0 + = + =JJG JJG JJG G JJJG JJG G (1) ( ) ( )JA JB 3JC 0 JA JA AB 3 JA AC 0 AB JA 3AC 0+ = + + + = =JJG JJG JJG G JJG JJG JJJG JJG JJJG G JJJG JJG JJJG G (2) Cong (1) va (2) ta c IJ 2AC 0 hay IJ 2AC = =JJG JJJG G JJG JJJG Vay IJ va AC cung phng. 4) Ta co: 2AI AR AS= +JJG JJJG JJJG AR AB AP AM= + JJJG JJJG JJJG JJJJG (v BP va MR co cung trung iem) AS AC AQ AM= + JJJG JJJG JJJG JJJJG (v CQ va MS co cung trung iem) Vay 2AI AR AS AB AC AP AQ 2AM AB AC AM= + = + + + = + JJG JJJG JJJG JJJG JJJG JJJG JJJG JJJJG JJJG JJJG JJJJG . V M tren oan BC nen: ( )AM kAB 1 k AC, 0 k 1= + JJJJG JJJG JJJG . Do o ( ) ( ) 1 12AI 1 k AB kAC AI 1 k . .AB k. AC

2 2= + = +JJG JJJG JJJG JJG JJJG JJJG .

Goi E, F la trung iem AB va AC th ( )AI 1 k AE kAF= + JJG JJJG JJJG I thuoc oan EF. Vay tap hp I la ng trung bnh cua tam giac ABC, ng vi canh BC. 5) Goi M, H, P la cac iem xac nh bi: 3PA PB 3HA 2HC 2MC MB = + = JJJG JJJG JJJG JJJG JJJG JJJG . Do A, B, C khong thang hang nen NA cat PC tai N va 3NA NB 2NC 0 + =JJJG JJJG JJJG G nen ta co: 3NA NM 0+ =JJJG JJJG G hay A, M, N thang hang M BC M AN BC = . Tng t H NB CA, P NC AB= = . 6) Theo tnh chat phan giac, ta co: LB AB 3 DB AB 3va 2LB LK 0

LK BK 2 DC AC 4= = = = + =JJJG JJJG G

2LB 3LK 0 2AB 3AK 5AL3AC 6AK 7AD 10AL

4DB 3DC 0 4AB 3AC 7ADAD 10

0 7AD 10ALAL 7

+ = + = = + = + = = =

JJJG JJJG G JJJG JJJG JJJG JJJG JJJG JJJG JJJGJJJG JJJG G JJJG JJJG JJJGJJJG JJJG

Bai 2: Cho tam giac ABC, at AB u, AC v= =JJJG JG JJJG G . 1) Goi P la iem oi xng cua B qua C. Tnh AP theo u, v

JJJG JG G.


30

2) Goi Q va R la iem nh bi 1 1AQ AC va AR AB2 3

= =JJJG JJJG JJJG JJJG . Tnh RP va RQ theo u va vJJJG JJJG JG G . T o suy ra ba iem P, Q, R thang hang.

3) Neu E, F xac nh bi ( )1 1AE AB, AF AC k 0; k 1k k 1


. Chng minh EF luon i qua mot iem

co nh khi k thay oi. Hng dan: 1) Xet tam giac ABP, ta co: 2AC AB AP AP 2AC AB 2v u= + = = JJJG JJJG JJJG JJJG JJJG JJJG G JG . Vay AP 2v u= JJJG G JG . 2) Ta co: 1RP RA AP AB AP

3= + = +JJJG JJJG JJJG JJJG JJJG

Theo cau 1: ( )1 4AP 2v u RP u 2v u hay RP 3v 2u3 6= = + = JJJG G JG JJJG JG G JG JJJG G JG ( )1 1 1 1 1RQ RA AQ AB AC u v 3v 2u3 2 3 2 6= + = + = + = JJJG JJJG JJJG JJJG JJJG JG G G JG Ta co:

( )( )

4RP 3v 2u

6 RP 4RQ RP va RQ cung phng hay R, P, Q thang hang.1RQ 3v 2u6

= = =

JJJG G JGJJJG JJJG

JJJG G JG

3) Ta co:

( ) ( ) ( )1AE AB AB kAEk AC AB 1 k AF kAE BC 1 k AF kAE

1 AC 1 k AFAF ACk 1

= = = + = + = + = +

JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJGJJJG JJJGJJJG JJJG (1)

Goi M la iem thuoc EF sao cho ( )k 1 MF kME 0+ =JJJG JJJG G (2) Va (1) ta co: BC AM= JJJG JJJJG M co nh va M la nh th t cua hnh bnh hanh ABCM. T ( ) ( ) k2 k 1 MF kME MF ME

k 1 + = = +


Vay M, E, F thang hang hay EF i qua iem co nh M. Bai 3: Cho tam giac ABC, trong tam G. 1) Bieu dien JI theo c AB, b AC= =JJG G JJJG G JJJG . 2) Chng minh IJ qua G ( )IJ cung phng IGJJG JJG . 3) Lay iem M, N ,P sao cho MB 2MC NA 2NC PA PB 0 = + = + =JJJG JJJG JJJG JJJG JJJG JJJG G .

a) Tnh PM, PN theo AB va ACJJJG JJJG JJJG JJJG

. b) Chng minh rang M, N, P thang hang.

Hng dan: 1) ( )IA 2IB IA 2 IA AB IA 2AB hay AI 2c= = + = =JJG JJG JJG JJG JJJG JJG JJJG JJG G


31

( ) 2 23JA 2JB 0 3JA 2 JA AB 0 AJ AC b5 5+ = + + = = =JJG JJG G JJG JJG JJJG G JJG JJJG G Vay 2JI JA AI AI AJ 2c b

5= + = = JJG JJG JJG JJG JJG G G

2) ( ) ( )1 1 6IG AG AI AB AC AI b 5c JI IG IJ // IG hay IJ qua G.3 3 5= = + = = JJG JJJG JJG JJJG JJJG JJG G G JJG JJG JJG JJG 3) a) Ta co: 1 2PN PA AN AB AC

2 3= + = +JJJG JJJG JJJG JJJG JJJG

( )1 1 3PM PB BM AB 2BC AB 2 BA AC AB 2AC2 2 2= + = + = + + = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG b) Theo cau 3a, ta co: PM 3PN=JJJG JJJG . Vay P, M, N thang hang. Bai 4: Cho tam giac ABC. 1) MB 3MC, NA 3NC 0, PA PB 0= + = + =JJJG JJJG JJJG JJJG G JJJG JJJG G . Phan tch PM, PN theo a BC, b CA= =JJJG JJJG G JJJG G JJJG . Chng minh M, N, P thang hang. 2) Xac nh hai iem I, F bi IA 3IC 0, FA 2FB 3FC 0+ = + + =JJG JJG G JJJG JJG JJJG G . Chng minh I, F, B thang hang. Hng dan:

1) Cach 1: ( ) ( )1 3 1 3 1 3 1PM PB BM AB BC CB CA BC hay PM a b a a b2 2 2 2 2 2 2= + = + = = + = JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG G G G G G (1). ( )1 3 1 3 2a bPN PA AN AB CA a b b2 4 2 4 4+= + = = + =

G GJJJG JJJG JJJG JJJG JJJG G G G (2)

T (1), (2) suy ra PM 2PN PM // PN= JJJG JJJG JJJG JJJG P, M, N thang hang. Cach 2: Ta co:

( )

1AP AB2PA PB 03

NA 3NC 0 AN AC4

MB 3MC 3 1AB AM 3 AC AM AM AC AB2 2

= + = + = = = = =

JJJG JJJGJJJG JJJG GJJJG JJJG G JJJG JJJGJJJG JJJG

JJJG JJJJG JJJG JJJJG JJJJG JJJG JJJG

Va

3MP AP AM AB AC2 MP 2MN

1 3MN AN AM AB AC2 4

= = = = =

JJJG JJJG JJJJG JJJG JJJGJJJG JJJG

JJJG JJJG JJJJG JJJG JJJG M, N, P thang hang.

2) T IA 3IC 0+ =JJG JJG G ta co FA 3FC 4FI+ =JJJG JJJG JJG Va FA 2FB 3FC 0 2FB 4FI 0+ + = + = JJJG JJG JJJG G JJG JJG G F, B, I thang hang.

Bai 5: Cho hnh bnh hanh ABCD, M va N la hai iem lan lt tren oan AB va CD sao cho AM 1AB 3

= va CN 1CD 2

= .


32

1) Tnh AN theo AB, ACJJJG JJJG JJJG

. 2) Goi G la trong tam tam giac MNB, tnh AG theo AB, AC

JJJG JJJG JJJG.

3) Goi I la iem nh bi BI kBC. Tnh AI theo AB, AC va k. Tnh k e AI qua G=JJG JJJG JJG JJJG JJJG . Hng dan: 1) Ta co:

AD AC 2AN 12AN 2AC AB AN AC AB

2AD DC AC

+ = = = + =

JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJGJJJG JJJG JJJG

2) AB AM AN 3AG+ + =JJJG JJJJG JJJG JJJG (tnh chat trong tam) 1 1 13AG AB AN AB AB AC AB AC3 2 3

= + + = + +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG

5 1

AG AB AC18 3

= +JJJG JJJG JJJG 3) ( ) ( )AI AB BI AB kBC AB k BA AC 1 k AB kAC= + = + = + + = +JJG JJJG JJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG ieu kien can va u e AI i qua G la AI, AG

JJG JJJG cung phng ( )5 1 6k 1 k k

18 3 11 = = .

Bai 6: Cho tam giac ABC. Tren BC lay iem D sao cho: 3

BD BC5

=JJJG JJJG (1). Goi E la mot iem thoa 4EA 2EB 3EC 0+ + =JJJG JJJG JJJG G (2). 1) Tnh ED theo EB va EC

JJJG JJJG JJJG.

2) Chng minh A, E, D thang hang. 3) Tren AC lay iem F sao cho AF xAC=JJJG JJJG . Hay xac nh x sao cho B, E, F thang hang. Hng dan:

1) T (1) ta co: ( )3 3 2 3 2BE ED BE EC ED EC BE EC EB5 5 5 5 5+ = + = = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG . 2) T (2) ta co: 4EA 2EB 3EC 4EA 5ED 0+ + = + =JJJG JJJG JJJG JJJG JJJG G . Suy ra

5EA ED

4= JJJG JJJG hay A, E, D thang hang.

3) T (2) ta co: ( )1EB 4EA 3EC2= +JJJG JJJG JJJG (1) ( ) ( )AF xAC AE EF x AE EC EF 1 x EA xEC= + = + = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG (2) Ba iem B, E, F thang hang ( ) ( ) ( )( )1 x x 3EB kEF k 0 do 1 , 2 x32 7

2

= = = JJJG JJJG

.

Bai 7: Cho tam giac ABC, M la iem sao cho BM BC 2AB= JJJG JJJG JJJG (1) va N la iem xac nh bi CN xAC BC= JJJG JJJG JJJG (2). 1) Xac nh x sao cho A, M, N thang hang. 2) Xac nh x e MN i qua trung iem cua canh BC. Hng dan:


33

1) ( )AM AB BM AB BC 2AB BC BA= + = + = +JJJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG ( ) ( ) ( )AN AC CN AC xAC BC x 1 AC BC 1 x AB xBC= + = + = + = + +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG A, M, N thang hang khi ( ) 1AM kAN k 0 x

2= = JJJJG JJJG

2) D la trung iem BC: 1MD 2AB BC2

= JJJG JJJG JJJG ( ) ( )MN MA AN 2 x AB x 1 BC= + = + + JJJG JJJG JJJG JJJG JJJG e M, N, D thang hang, ieu kien can va u la

2MN kMD x

5= =JJJG JJJG

Bai 8: Cho tam giac ABC, k la hang so, M la iem di dong sao cho v MN 2MA 3MB kMC= = +G JJJG JJJG JJJG JJJG . 7) Vi k 1= . Chng minh MNJJJG co phng khong oi. 8) Vi k 1 . Chng minh MN luon i qua mot iem co nh.

Hng dan: 1) ( ) ( )k 1 : v 2MA 3MB MC 2MA 3 MA AB MA AC 3AB AC= = + = + + + = +G JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG vect khong oi. Vay MN

JJJG co phng khong oi.

2) Goi I: ( ) ( ) ( )2IA 3IB kIC 0 2 CA CI 3 CB CI kCI 0 1 k CI 3CB 2CA + = = = JJG JJG JJG G JJJG JJG JJJG JJG JJG G JJG JJJG JJJG 3CB 2CACI I co nh, k 1

1 k = .

Va: ( ) ( ) ( ) ( ) ( )MN 2 MI IA 3 MI IB k MI IC k 1 MI 2IA 3IB kIC k 1 MI= + + + + = + + = JJJG JJJG JJG JJJG JJG JJJG JJG JJJG JJG JJG JJG JJJG . Vay ( )MN k 1 MI MN // MI MN i qua I co nh= JJJG JJJG .

Bai 9: Cho tam giac ABC, trong tam G. Cac iem M, N thoa man: 1

3MA 4MB 0,CN BC2

+ = =JJJG JJJG G JJJG JJJG . Chng minh MN i qua trong tam G cua tam giac ABC.

Hng dan: 3

3MA 4MB 0 MB AB7

1 3CN BC BN BC

2 2

+ = =

= =

JJJG JJJG G JJJG JJJG

JJJG JJJG JJJG JJJG

Ta co: 3 3

MN MB BN AB BC7 2

= + = +JJJG JJJG JJJG JJJG JJJG (1) G la trong tam tam giac ABC nen :

( )3 2 3 2 1 2 1MG MB BG AB BE AB . BA BC AB BC7 3 7 3 2 21 3= + = + = + + = +JJJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG (2) T (1) va (2) suy ra

9MN MG

2=JJJG JJJJG .


34

Vay ba iem M, N, G thang hang. Bai 3: Cho tam giac ABC. 6) MN v 2MA 3MB kMC= = + +JJJG G JJJG JJJG JJJG .

a. Khi k 5 . Chng minh gia cua MNJJJG luon qua mot iem co nh. b. Tm k e MN

JJJG la mot vect khong oi.

7) Lay E, F tren tam giac ABC sao cho ( )1 1AE AB, AF AC k 0; 1k k 1


. Chng minh EF luon qua

mot iem co nh. Hng dan:

6) a) Goi I la iem thoa ( ) ( )2IA 3IB kIC 0 2IA 3 IA AB k IA AC 0+ + = + + + + =JJG JJG JJG G JJG JJG JJJG JJG JJJG G ( ) ( )3AB kAC5 k IA 3AB kAC IA k 5

5 k + = = +JJJG JJJGJJG JJJG JJJG JJG

I co nh.

( ) ( ) ( ) ( )MN v 2 MI IA 3 MI IB k MI IC 5 k MI MN // MI= = + + + + + = + JJJG G JJJG JJG JJJG JJG JJJG JJG JJJG JJJG JJJG hay MN qua I co nh. b) ( ) ( ) ( )MN 2MA 3 MA AB k MA AC 5 k MA 3AB kAC= + + + + = + + +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG MN

JJJG khong oi khi 5 k 0 k 5+ = = .

7) ( ) ( )AC k 1 AF kAF AF k AE EF AF AC AB kEF AF= + = + = + + = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG Hay kEF AC AB AF BC AF= = JJJG JJJG JJJG JJJG JJJG JJJG Ve AI BC=JJG JJJG th AI AF kEF kEF FI = = JJG JJJG JJJG JJJG JJG E, F, I thang hang EF qua I co nh. Bai 10: Cho hnh bnh hanh ABCD. 6) Goi I, F, K la cac iem xac nh bi AI AB, AF AC, AK AD= = = JJG JJJG JJJG JJJG JJJG JJJG . Chng minh ieu kien can va u e I, F, K thang hang la ( )1 1 1 , , 0= + . 7) Goi M, N la 2 iem lan lt tren oan AB va CD sao cho AM 1 CN 1,

AB 3 CD 2= = . Goi G la trong tam tam giac

BMN. a) Tnh AN, AG theo AB, AC


b) Goi H la iem nh bi BH kBC=JJJG JJJG . Tnh AH theo AB, AC va kJJJG JJJG JJJG . Tm k e AH qua G. Hng dan:

6) Ta co: KI AI AK AB AD= = JJG JJG JJJG JJJG JJJG KF AF AK AC AD= = JJJG JJJG JJJG JJJG JJJG Ma AC AB AD= +JJJG JJJG JJJG Vay ( )KF AB AD= + JJJG JJJG JJJG ieu kien can va u e K, I, F thang hang va ton tai so k sao cho KF kKI=JJJG JJG ( ) ( ) ( )AB AD k AB k AD k AB k AD 0 + = + + =JJJG JJJG JJJG JJJG JJJG JJJG G (1)


35

Do AB, ADJJJG JJJG

khong cung phng nen ( )1 k 0 k k 1 = = + = = = ( )1 1 1 , , 0 = + (pcm). 7) a) Ta co:

AD AC 2AN 1AC AB 2AN AC AN AC AB

2AD AB AC

+ = = = + =

JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJGJJJG JJJG JJJG

Va ta co: 1

3AG AB AM AN AB AN AB3

= + + = + +JJJG JJJG JJJJG JJJG JJJG JJJG JJJG

4 1 53AG AB AC AB AB AC3 2 6

5 1AG AB AC18 3

= + = +

= +


JJJG JJJG JJJG

b) ( )AH AB BH AB kBC AB kBA kAC 1 k AB kAC= + = + = + + = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG ieu kien can va u e AG qua H la AGJJJG cung phng AHJJJG ( )5 1 6k 1 k k

18 3 11 = = .

Bai 11: Cho tam giac ABC. 11) Goi I la trung iem cua BC, D va E la hai iem sao cho BD DE EC= =JJJG JJJG JJJG .

c) Chng minh rang AB AC AD AE+ = +JJJG JJJG JJJG JJJG . d) Tnh AS AB AD AC AE theo AI= + + +JJJG JJJG JJJG JJJG JJJG JJG , suy ra A, I , S thang hang.

12) Goi M la iem nh bi BM BC 2AB= JJJG JJJG JJJG , N la iem nh bi CN xAC BC= JJJG JJJG JJJG . c) Xac nh x e A, M, N thang hang.

d) Xac nh x e MN qua iem gia I cua BC. Tnh IMIN

.

Hng dan: 1) a) Ta co:

( )( )

AB AC 2AI I la tr ung iem BCAB AC AD AE

AD AE 2AI I la tr ung iem DE

+ = + = ++ =

JJJG JJJG JJGJJJG JJJG JJJG JJJG

JJJG JJJG JJG

b) ( ) ( )AS AB AC AD AE 2AI 2AI 4AI= + + + = + =JJJG JJJG JJJG JJJG JJJG JJG JJG JJG V AS 4AI=JJJG JJG nen AS va AIJJJG JJG cung phng hay A, S, I thang hang. 2) a) Ta co: BM BC 2AB 2AB BC BM= = JJJG JJJG JJJG JJJG JJJG JJJG ( ) ( )Hay 2AB BA AC BA AM AC AM AM AC 2AB= + + = = JJJG JJJG JJJG JJJG JJJJG JJJG JJJJG JJJJG JJJG JJJG (1) ( ) ( )CN xAC BC xAC CN CB CA AN CA AB= = = + +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG Hay xAC AN AB AN xAC AB= = +JJJG JJJG JJJG JJJG JJJG JJJG (2) e A, M, N thang hang AM, AN JJJJG JJJG cung phng AM kAN =JJJJG JJJG


36

Chon ( ) ( )k 2 th AM 2AN AM 2AN 0 AC 2AB 2 xAC AB 0= = + = + + =JJJJG JJJG JJJJG JJJG G JJJG JJJG JJJG JJJG G ( ) 12x 1 AC 0 2x 1 0 x

2 + = + = = JJJG G

Vay vi 1

x2

= th A, M, N thang hang. b) ( ) ( ) ( )BM BC 2AB 2 BI IA BI IC BI IM 0= + + + + =JJJG JJJG JJJG JJG JJG JJG JJG JJG JJJG G 2IB 2IA IC IM IM 2IA 3IC = + = +JJG JJG JJG JJJG JJJG JJG JJG ( )CN xAC BC x CI IA CI IN CI IB 0= + + + =JJJG JJJG JJJG JJG JJG JJG JJG JJG JJG G ( )xCI xIA IN IB IN xIA x 1 IC = + = + JJG JJG JJG JJG JJG JJG JJG e M, N, I thang hang IM va IN cung phng IM kIN =JJJG JJG JJJG JJG . Chon ( )k 5 th IM 5IN IM 5IN 0 2IA 3IC 5 xIA x 1 IC 0 = = + = + + =

JJJG JJG JJJG JJG G JJG JJG JJG JJG G

( ) 22 x 1 3x 0 x5

+ = =

Vay vi 2

x5

= th M, N, I thang hang.

IM 2IA 3IC2 IMKhi x th IM 5IN 52 35 ININ IA IC

5 5

= += = = =

JJJG JJG JJGJJJG JJG

JJG JJG JJG

Bai 12: Cho ba iem co nh A, B, C va ba so thc sao cho a b c 0+ + . 6) Chng minh rang ch co mot iem G thoa man he thc aGA bGB cGC 0+ + =JJJG JJJG JJJG G . 7) Goi M va P la hai iem di ong sao cho MP aMA bMB cMC= + +JJJG JJJG JJJG JJJG . Chng minh rang ba iem G, M, P thang hang.

Hng dan: 7) Ta co: ( ) ( )aGA bGB cGC 0 aGA b GA AB c GA AC 0+ + = + + + + =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG G ( ) bAB cACa b c GA bAB cAC AG

a b c+ + + = = + +

JJJG JJJGJJJG JJJG JJJG JJJG khong oi.

Do o G c xac nh va duy nhat. 8) ( ) ( ) ( )MP aMA bMB cMC aMP a MG GA b MG GB c MG GC= + + = = + + + + +JJJG JJJG JJJG JJJG JJJG JJJJG JJJG JJJJG JJJG JJJJG JJJG ( ) ( ) ( )MP a b c MG aGA bGB cGC a b c MG= + + + + + = + +JJJG JJJJG JJJG JJJG JJJG JJJJG . Vay M, P ,G thang hang. Bai 13: Cho tam giac ABC. 7) Goi M la trung iem BC, I va J la hai iem xac nh bi AI AB va AJ AC= = JJG JJJG JJG JJJG . Tm he thc lien he gia va e AM cat IJ tai trung iem cua AM.


37

8) Goi P la iem lu ong. Dng vect PQ 2PA 3PB PC= + JJJG JJJG JJJG JJJG . Chng minh PQ i qua mot iem co nh khi P thay oi . Goi H la trung iem cua CQ. Chng minh rang PH i qua mot iem co nh khi P thay oi.

Hng dan: a. Neu 0 0 = = : khong thoa man gia thiet. Do vay 0 va 0 .

Ta co: ( )1 1 1 1 1 1AM AB AC AI AI AI AI2 2 2 2 = + = + = + JJJJG JJJG JJJG JJG JJG JJG JJG

Do o AM se cat IJ tai K ma 1 1AM AK2 2

= + JJJJG JJJG

(tam t c)

Vay K la trung iem cua AM th 1 1 1 1

2 42 2

+ = + = b. Goi E la iem nh bi ( ) ( )2EA 3EB EC 0 2EA 3 EA AB EA AC 0+ = + + + =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG G

5AE 3AB AC E co nh = JJJG JJJG JJJG . Ta chng minh PQ qua E. ( ) ( ) ( )PQ 2PA 3PB PC 2 PE EA 3 PE EB PE EC 4PE 2EA 3EB EC 4PE= + = + + + + = + + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG P, Q, E thang hang PQ qua iem co nh E. Ta co: ( ) ( ) ( )1 1 1 5PH PQ PC 2PA 3PB 2 3 PJ PJ2 2 2 2= + = + = + =JJJG JJJG JJJG JJJG JJJG JJG JJG P, H, J thang hang PH qua iem co nh J. Bai 8: Cho bon iem A, B, C, M thoa man MA 2MB 3MC 0+ =JJJG JJJG JJJG G . Chng minh A, B, C thang hang.

Hng dan:

( ) ( ) 3MA 2MB 3MC 0 MA 2 MA AB 3 MA AC 0 2AB 3AC 0 AB AC2+ = + + + = = =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG JJJG G JJJG JJJG G JJJG JJJG A, B, C thang hang. Bai 14: Cho tam giac ABC, M va N thay oi sao cho MN 2MA 3MB MC= + JJJG JJJG JJJG JJJG 3) Tm iem I thoa man 2IA 3IB IC 0+ =JJG JJG JJG G . 4) Chng minh ng thang MN luon i qua mot iem co nh.

Hng dan: 3) ( ) ( )0 2IA 3IB IC 2IA 3 IA AB IA AC 4IA 3AB AC= + = + + + = + G JJG JJG JJG JJG JJG JJJG JJG JJJG JJG JJJG JJJG ( )1IA AC 3AB4 = JJG JJJG JJJG : ang thc xac nh iem I 4) ( ) ( ) ( ) ( )MN 2MA 3MB MC 2 MI IA 3 MI IB MI IC 4MI 2IA 3IB IC 4MI= + = + + + + = + + =JJJG JJJG JJJG JJJG JJJG JJG JJJG JJG JJJG JJG JJJG JJG JJG JJG JJJG M, N, I thang hang hay MN i qua iem I co nh. Bai 15: Cho tam giac ABC, goi I, J la hai iem xac nh bi IA 2IB; 3JA 2JC 0= + =JJG JJG JJG JJG G 3) Tnh IJ theo AB va AC

JJG JJJG JJJG.

4) Chng minh ng thang IJ i qua trong tam G cua tam giac ABC.


38

Hng dan: 3) IJ AJ AI= JJG JJG JJG ( )AI 2BI 2 BA AI AI 2AB= = + =JJG JJG JJJG JJG JJG JJJG ( ) 23JA 2JC 0 3JA 2 JA AC 0 AJ AC5+ = + + = =JJG JJG G JJG JJG JJJG G JJG JJJG Vay

2IJ AC 2AB

5= JJG JJJG JJJG

4) Ta co: ( )1AG AB AC3= +JJJG JJJG JJJG Mat khac: ( )1IG AG AI AC 5AB3= = +JJG JJJG JJG JJJG JJJG 6IJ IG

5 = I, J, G thang hang.

Bai 16: Cho tam giac ABC, hai iem M, N thay oi sao cho MN 2MA MB MC= +JJJG JJJG JJJG JJJG 4) Dng iem I sao cho 2IA IB 3IC 0 + =JJG JJG JJG G . 5) Chng minh rang MN i qua mot iem co nh khi M thay oi. 6) Goi P la trung iem cua BN. Chng minh MP i qua mot iem co nh khi M thay oi.

Hng dan: 4) ( ) ( ) ( )2IA IB 3IC 0 2 OA OI OB OI 3 OC OI 0 2OA OB 3OC 4OI + = + = + =JJG JJG JJG G JJJG JJG JJJG JJG JJJG JJG G JJJG JJJG JJJG JJG Neu chon O C th 4CI 2CA CB = JJG JJJG JJJG : ang thc nay hoan toan xac nh I. 5) ( ) ( ) ( )MN 2MA MB 3MC 2 MI IA MI IB 3 MI IC MN 4MI= + = + + + + =JJJG JJJG JJJG JJJG JJJG JJG JJJG JJG JJJG JJG JJJG JJJG M, N, I thang hang hay MN qua I co nh. 6) ( ) ( )1 1MP MB MN 2MA 3MC2 2= + = +JJJG JJJG JJJG JJJG JJJG Goi J la iem thoa man 2JA 3JC 0+ =JJG JJG G , J xac nh. ( ) ( )1 5MP 2 MJ JA 3 MJ JC MJ2 2 = + + + = JJJG JJJG JJG JJJG JJG JJJG . Vay M, P, J thang hang hay MP qua J co nh.

VAN E 5 TAP HP QUY TCH

Ta chon mot hoac hai iem co nh A, B. ta bien oi he thc ve mot trong cac dang sau ay: 1) Neu AM

JJJJG cung phng vect co nh a

G (hay a

G co phng khong oi) th tap hp cac iem M la ng

thang qua A va cung phng vi aG

. 2) Neu AM R=JJJJG (hang so) th tap hp cac iem M la ng tron tam A, ban knh R.


39

3) Neu mAM nBM=JJJJG JJJG , m, n la hai hang so, m 0 th tap hp cac iem M la ng tron ng knh CD vi C va D la hai iem chia trong va ngoai oan AB theo t so

mk

n= .

ac biet khi m n= th tap hp cac iem M la ng trung trc cua oan AB. Bai 1: Cho tam giac ABC. 1) M la iem di ong. Dng MN 2MA 3MB MC= + JJJG JJJG JJJG JJJG . Chng minh MN luon i qua mot iem co nh khi M thay oi. 2) Goi P la trung iem cua CN. Chng minh rang MP luon luon qua mot iem co nh khi M thay oi. 3) Keo dai AB mot oan BE AB= , goi F la trung iem AC. Ve hnh bnh hanh EAFG. ng thang AG cat BC tai K. Tnh KB

KC

JJJGJJJG .

Hng dan: 1) Goi I la iem co nh, xac nh bi 2IA 3IB IC 0+ =JJG JJG JJG G . Ta co: ( )MN 2MA 3MB MC 2 3 1 MI 4MI= + = + =JJJG JJJG JJJG JJJG JJJG JJJG Vay M, N, I thang hang hay MN luon qua iem co nh I.

2) V P la trung iem cua CN nen ta co: ( ) ( )1 1MP MN MC 2MA 3MB2 2= + = +JJJG JJJG JJJG JJJG JJJG . Goi J la iem co nh xac nh bi 2JA 3JB 0+ =JJG JJG G . Ta co: ( ) ( )1 1 5MP 2MA 3MB 2 3 MJ MJ2 2 2= + = + =JJJG JJJG JJJG JJJG JJJG . Vay MP luon qua iem co nh J.

3) Ta co: 1AG AE AF 2AB AC2

= + = +JJJG JJJG JJJG JJJG JJJG

Vay AG cat BC tai K ma 1 KB 12KB KC 02 4KC

+ = =JJJGJJJG JJJG G JJJG .

Bai 2: Cho tam giac ABC. Tm tap hp cac iem M sao cho: (vi k R ) 1) MA kMB kMC 0+ + =JJJG JJJG JJJG G . 2) ( )kMA 1 k MB 0+ =JJJG JJJG G . 3) ( )2MA 3 k MB kMC 0+ + =JJJG JJJG JJJG G . 4) v MA MB 2MC cung phng BC= + +G JJJG JJJG JJJG JJJG . 5) ( )2MA 1 k MB 3kMC 0 + =JJJG JJJG JJJG G . 6) 3MA MB MC MB MC

2+ + = +JJJG JJJG JJJG JJJG JJJG .

7) MA BC MA MB+ = JJJG JJJG JJJG JJJG . 8) 2MA MB 4MA MC+ = JJJG JJJG JJJG JJJG .


40

9) 4MA MB MC 2MA MB MC+ + = JJJG JJJG JJJG JJJG JJJG JJJG . 10) MA MB MA MB+ = +JJJG JJJG JJJG JJJG .

Hng dan:

1) Neu 1k2

, chon iem I xac nh bi IA kIB kIC 0+ + =JJG JJG JJG G Ta co: ( )MA kMB kMC 0 1 2k MI 0 M I+ + = + = JJJG JJJG JJJG G JJJG G Vay tap hp can tm gom mot iem I.

Neu 1

k2

=

Ta co: ( ) ( )1 1MA MB MC 0 MA MB MA MC 02 2 = + =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG G BA CA 0 + =JJJG JJJG G (vo ly) Vay tap iem can tm la tap rong. 2) Chon iem I thoa: ( )kIA 1 k IB 0+ =JJG JJG G Ta co: ( )kMA 1 k MB 0 MI 0+ = =JJJG JJJG G JJJG G Vay tap hp can tm gom mot iem I. 3) Tap can tm la mot iem I thoa ( )2IA 3 k IB kIC 0+ + =JJG JJG JJG G 4) Chon I la tam t c cua ( ) ( ) ( )A 1 , B 1 ,C 2 Ta co: v MA MB 2MC 4MI. Vay v cung phng BC MI cung phng BC= + + = G JJJG JJJG JJJG JJJG G JJJG JJJG JJJG M thuoc ng thang qua I va song song vi BC. 5) Neu 1k

4 , chon I la tam t c cua ( ) ( ) ( )A 2 , B k 1 ,C 3k .

Tap iem I can tm ch gom mot iem I.

Neu 1k4

=

Ta co: ( ) ( )5 32MA MB MC 0 5 MA MB 3 MA MC 0 5BA 3CA 04 4 = + = + =JJJG JJJG JJJG G JJJG JJJG JJJG JJJG G JJJG JJJG G (vo ly) Vay tap can tm la tap rong. 6) Goi G la trong tam tam giac ABC, I la trung iem BC

Ta co: 3

3MG 2MI MG MI2

= = JJJJG JJJG JJJJG JJJG M thuoc ng trung trc cua IG. 7) Dng hnh bnh hanh ABCD. Ta co: MA BC MA MB MA AD BA MD BA DM BA+ = + = = =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG Vay M thuoc ng tron tam D, ban knh R AB= . 8) Chon P la tam t c cua ( ) ( )A 2 , B 1 , Q la tam t c cua ( ) ( )B 4 ,C 1 . Ta co: 2MA MB 4MB MC 3MP 3MQ MP MQ+ = = =JJJG JJJG JJJG JJJG JJJG JJJJG Vay M thuoc trung trc cua PQ. 9) Chon E la tam t c cua ( ) ( ) ( )A 4 , B 1 ,C 1


41

Ta co: 4MA MB MC 2MA MB MC 6ME MA MB MA MC+ + = = + JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG

16ME BA CA ME BA CA

6 = + = +JJJG JJJG JJJG JJJG JJJG .

Vay M thuoc ng tron tam E, ban knh 1

R BA CA6

= +JJJG JJJG 10) Dng hnh bnh hanh MANB. Ta co: MA MB MN+ =JJJG JJJG JJJG Lai co: MN MB BN + . Dau = xay ra khi B thuoc MN. Ma MA MB MA MB MN MA MB MN MB BN B MN+ = + = + = + JJJG JJJG JJJG JJJG JJJG JJJG JJJG . Vay tap cac iem M la ng thang AB tr tat ca cac iem trong oan AB. Bai 3: Cho hnh bnh hanh ABCD. 1) M va N la hai iem lu ong xac nh bi MN 3MA 2MB 2MC MD= +JJJG JJJG JJJG JJJG JJJG .

a) Chng minh MNJJJG

la vect khong oi. Tm tap hp iem M biet rang gia cua MNJJJG

qua tam O cua hnh bnh hanh.

b) Tm tap hp iem M khi N chuyen ong tren AC. 2) Goi ABCD la hnh bnh hanh. Cac iem P va Q nh bi: ( )PC kPC 0,QD kQD 0 k 1 + = + = JJJG JJJG G JJJG JJJJG G , con M va N cua hnh bnh hanh ABCD nh bi MA kMA 0, NB kNB 0 + = + =JJJG JJJJG G JJJG JJJG G .

a) Chng minh MNPQ la hnh bnh hanh. b) Tm tap hp tam hnh bnh hanh khi M chay tren AA

Hng dan: 1) a) ( ) ( ) ( )MN 2 MA MB MA MC MD MC 2BA CA CD= + + = + +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG MN 2BA CA CD = + +JJJG JJJG JJJG la vect khong oi. Tap cac iem M chnh la ng thang qua O va song song vi vect khong oi 2BA CA CD+ +JJJG JJJG JJJG . b) Ta dng mot iem ac biet cua tap can tm. Dng A0 sao cho: 0A A 2BA CA CD= + +

JJJJG JJJG JJJG JJJG. Khi o ta co: 0MN A A=

JJJG JJJJG.

Vay tap iem can tm la ng thang qua A0 va song song AC. 2) a) ( )MN MA AB BN va kMN k MA A B B N = + + = + +JJJG JJJG JJJG JJJG JJJG JJJJG JJJJG JJJJG

( ) ( ) ( ) ( )( )k 1 MN MA kMA AB kA B NB kNB

k 1 MN AB kA B

+ = + + + + + = +

JJJG JJJG JJJJG JJJG JJJJG JJJG JJJGJJJG JJJG JJJJG

Tng t ta co: ( )k 1 QP DC kD C + = +JJJG JJJG JJJJG Ma AB DC, A B D C = =JJJG JJJG JJJJG JJJJG Nen ( ) ( )k 1 MN k 1 QP MN QP+ = + =JJJG JJJG JJJG JJJG . Vay MNPQ la hnh bnh hanh. b) Goi E la tam cua hnh bnh hanh MNPQ, ta co: ( )MA NB PC QD k MA NB PC QD 0 4EO 4kEO 0 EO kEO 0 + + + + + + + = + = + =JJJG JJJG JJJG JJJG JJJJG JJJG JJJG JJJJG G JJJG JJJG G JJJG JJJG G


42

E, O, O thang hang, ong thi khi M chay tren oan AA, E chay tren oan OO trong o O la tam hnh bnh hanh ABCD, O la tam hnh bnh hanh ABCD. Bai 4: Cho ba iem co nh A, B,C khong thang hang. Goi M la iem di ong sao cho AM

JJJJG cung phng

BCJJJG

. 1) Tm tap hp cac iem M. 2) Goi N la iem sao cho ABMN la hnh bnh hanh. Tm tap hp cac iem N. 3) Goi I la tam cua hnh bnh hanh ABMN. Tm tap hp cac iem I.

Hng dan: 1) Tm tap hp cac iem M: Ta co: AM

JJJJG cung phng BC

JJJGBC kAM =JJJG JJJJG .

Vay tap hp iem M la ng thang qua A song song vi BC. 2) V ABMN la hnh bnh hanh AM cung phng BN. JJJJG JJJG Ma AM

JJJJG cung phng BC

JJJG

Nen BNJJJG

cung phng BCJJJG

B, N, C thang hang Vay tap hp iem N la ng thang BC ( )N BC . 3) Goi O la trung iem AB, suy ra O co nh. Theo tnh chat hnh bnh hanh ta co OI

JJG cung phng BC

JJJG.

Vay tap hp iem I la ng thang qua O va song song BC. Bai 5: Cho t giac ABCD. 1) Xac nh iem O sao cho OB 4OC 2OD+ =JJJG JJJG JJJG . 2) Tm tap hp cac iem M thoa he thc MB 4MC 2MD 3MA+ =JJJG JJJG JJJG JJJG .

Hng dan: 1) Ta co: ( ) ( )OB 4OC 2OD OB 4 OB BC 2 OB BD+ = + + = +JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG ( ) ( )3OB 2BD 4BC 2 BD BC 2BC 2CD 2BC 2 CD BC = = = = JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG 3OI 4CI=JJG JJG (I la trung iem BD)

4BO IC

3 =JJJG JJG

Vay O la nh hnh bnh hanh IBOE vi 4

IE IC3

=JJG JJG . 2) ( ) ( )MB 4MC 2MD 3MA MO OB 4 MO OC 2 MO OD 3MA+ = + + + + =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG

3MO OB 4OC 2OD 3MA

3MO 2OD 2OD 3MA

MO MA MO MA

+ + = + = = =

JJJG JJJG JJJG JJJG JJJGJJJG JJJG JJJG JJJGJJJG JJJG

Vay tap hp cac iem M la ng trung trc cua oan OA.


43

Bai 6: Cho luc giac eu ABCDEF tam O. Tm tap hp cac iem M thoa: MA MB MC MD ME MF 3 MA MD+ + + + + = JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG .

Hng dan: Ta co: MA MB MC MD ME MF 3 MA MD+ + + + + = JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG

MO OA MO OB MO OC MO OD MO OE MO OF 3 MO OA MO OD

16MO 3DA MO DA OA2

+ + + + + + + + + + + = + = = =

JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG

JJJG JJJG

M cach O co nh mot khoang khong oi OA. Vay tap hp iem M la ng tron tam O ban knh OA, ng tron ngoai tiep luc giac eu ABCDEF. Bai 7: Cho tam giac ABC, m thay oi, k la hang so. T

Documents

laisac.vecto