1. Cc nh ngha

( Vect l mt on thng c hng. K hiu vect c im u A, im cui B l .

( Gi ca vect l ng thng cha vect .

( di ca vect l khong cch gia im u v im cui ca vect, k hiu .

( Vect khng l vect c im u v im cui trng nhau, k hiu .

( Hai vect gl cng phng nu gi ca chng song song hoc trng nhau.

( Hai vect cng phng c th cng hng hoc ngc hng.

( Hai vect gl bng nhau nu chng cng hng v c cng di.

Ch : + Ta cn s dng k hiu biu din vect.

+ Qui c: Vect cng phng, cng hng vi mi vect.

Mi vect u bng nhau.2. Cc php ton trn vect

a) Tng ca hai vect

( Qui tc ba im: Vi ba im A, B, C tu , ta c: .

( Qui tc hnh bnh hnh: Vi ABCD l hnh bnh hnh, ta c: .

( Tnh cht:

;

;

b) Hiu ca hai vect

( Vect i ca l vect sao cho . K hiu vect i ca l .

( Vect i ca l .

( .

( Qui tc ba im: Vi ba im O, A, B tu , ta c: .

c) Tch ca mt vect vi mt s

( Cho vect v s k ( R. l mt vect c xc nh nh sau:

+ cng hng vi nu k ( 0, ngc hng vi nu k < 0.

+ .

( Tnh cht:

;

;

( k = 0 hoc .

( iu kin hai vect cng phng:

( iu kin ba im thng hng: A, B, C thng hng ( (k ( 0: .

( Biu th mt vect theo hai vect khng cng phng: Cho hai vect khng cng phng v tu . Khi (! m, n ( R: .

Ch :

( H thc trung im on thng:

M l trung im ca on thng AB ( ( (O tu ).

( H thc trng tm tam gic:

G l trng tm (ABC ( ( (O tu ).

VN 1: Khi nim vectBai 1. Cho t gic ABCD. C th xc nh c bao nhiu vect (khc ) c im u v im cui l cc im A, B, C, D ?Bai 2. Cho (ABC c A(, B(, C( ln lt l trung im ca cc cnh BC, CA, AB.

a) Chng minh: .

b) Tm cc vect bng .Bai 3. Cho t gic ABCD. Gi M, N, P, Q ln lt l trung im ca cc cnh AB, CD, AD, BC. Chng minh: .Bai 4. Cho hnh bnh hnh ABCD c O l giao im ca hai ng cho. Chng minh:

a) .

b) Nu th ABCD l hnh ch nht.Bai 5. Cho hai vc t . Trong trng hp no th ng thc sau ng: .Bai 6. Cho (ABC u cnh a. Tnh .Bai 7. Cho hnh vung ABCD cnh a. Tnh .Bai 8. Cho (ABC u cnh a, trc tm H. Tnh di ca cc vect . Bai 9. Cho hnh vung ABCD cnh a, tm O. Tnh di ca cc vect , , .VN 2: Chng minh ng thc vect Phn tch vect

chng minh mt ng thc vect hoc phn tch mt vect theo hai vect khng cng phng, ta thng s dng:

Qui tc ba im phn tch cc vect.

Cc h thc thng dng nh: h thc trung im, h thc trng tm tam gic.

Tnh cht ca cc hnh.Bai 1. Cho 6 im A, B, C, D, E, F. Chng minh:

a)

b) .Bai 2. Cho 4 im A, B, C, D. Gi I, J ln lt l trung im ca AB v CD. Chng minh:

a) Nu th

b) .

c) Gi G l trung im ca IJ. Chng minh: .

d) Gi P, Q ln lt l trung im ca AC v BD; M, N ln lt l trung im ca AD v BC . Chng minh cc on thng IJ, PQ, MN c chung trung im.Bai 3. Cho 4 im A, B, C, D. Gi I, J ln lt l trung im ca BC v CD. Chng minh: .Bai 4. Cho (ABC. Bn ngoi tam gic v cc hnh bnh hnh ABIJ, BCPQ, CARS. Chng minh: .Bai 5. Cho tam gic ABC, c AM l trung tuyn. I l trung im ca AM.

a) Chng minh: .

b) Vi im O bt k, chng minh: .Bai 6. Cho (ABC c M l trung im ca BC, G l trng tm, H l trc tm, O l tm ng trn ngoi tip. Chng minh:

a)

b)

c) .Bai 7. Cho hai tam gic ABC v A(B(C( ln lt c cc trng tm l G v G(.

a) Chng minh .

b) T suy ra iu kin cn v hai tam gic c cng trng tm.Bai 8. Cho tam gic ABC. Gi M l im trn cnh BC sao cho MB = 2MC. Chng minh: .Bai 9. Cho tam gic ABC. Gi M l trung im ca AB, D l trung im ca BC, N l im thuc AC sao cho . K l trung im ca MN. Chng minh:

a)

b) .Bai 10. Cho hnh thang OABC. M, N ln lt l trung im ca OB v OC. Chng minh rng:

a)

b)

c) .Bai 11. Cho (ABC. Gi M, N ln lt l trung im ca AB, AC. Chng minh rng:

a)

c)

c) .Bai 12. Cho (ABC c trng tm G. Gi H l im i xng ca B qua G.

a) Chng minh: v .

b) Gi M l trung im ca BC. Chng minh: .Bai 13. Cho hnh bnh hnh ABCD, t . Gi I l trung im ca CD, G l trng tm ca tam gic BCI. Phn tch cc vect theo .Bai 14. Cho lc gic u ABCDEF. Phn tch cc vect theo cc vect .Bai 15. Cho hnh thang OABC, AM l trung tuyn ca tam gic ABC. Hy phn tch vect theo cc vect .Bai 16. Cho (ABC. Trn cc ng thng BC, AC, AB ln lt ly cc im M, N, P sao cho .

a) Tnh theo

b) Chng minh: M, N, P thng hng.Bai 17. Cho (ABC. Gi A1, B1, C1 ln lt l trung im ca BC, CA, AB.

a) Chng minh:

b) t . Tnh theo .Bai 18. Cho (ABC. Gi I l im trn cnh BC sao cho 2CI = 3BI. Gi F l im trn cnh BC ko di sao cho 5FB = 2FC.

a) Tnh .

b) Gi G l trng tm (ABC. Tnh .Bai 19. Cho (ABC c trng tm G. Gi H l im i xng ca G qua B.

a) Chng minh: .

b) t . Tnh theo .VN 3: Xc nh mt im tho mn ng thc vect

xc nh mt im M ta cn phi ch r v tr ca im i vi hnh v. Thng thng ta bin i ng thc vect cho v dng , trong O v c xc nh. Ta thng s dng cc tnh cht v:

im chia on thng theo t s k.

Hnh bnh hnh.

Trung im ca on thng.

Trng tm tam gic, Bai 1. Cho (ABC . Hy xc nh im M tho mn iu kin: .Bai 2. Cho on thng AB c trung im I . M l im tu khng nm trn ng thng AB . Trn MI ko di, ly 1 im N sao cho IN = MI.

a) Chng minh: .

b) Tm cc im D, C sao cho: .

Bai 3. Cho hnh bnh hnh ABCD.

a) Chng minh rng: .

b) Xc nh im M tho mn iu kin: .

Bai 4. Cho t gic ABCD . Gi M, N ln lt l trung im ca AD, BC.

a) Chng minh: .

b) Xc nh im O sao cho: .Bai 5. Cho 4 im A, B, C, D. Gi M v N ln lt l trung im ca AB, CD, O l trung im ca MN. Chng minh rng vi im S bt k, ta c: .Bai 6. Cho (ABC. Hy xc nh cc im I, J, K, L tho cc ng thc sau:

a)

b)

c)

d) .Bai 7. Cho (ABC. Hy xc nh cc im I, J, K, L tho cc ng thc sau:

a)

b)

c)

d) .Bai 8. Cho (ABC. Hy xc nh cc im I, F, K, L tho cc ng thc sau:

a)

b)

c)

d) .Bai 9. Cho hnh bnh hnh ABCD c tm O. Hy xc nh cc im I, F, K tho cc ng thc sau:

a)

b)

c) .

Bai 10. Cho tam gic ABC v im M ty .

a) Hy xc nh cc im D, E, F sao cho , , . Chng minh D, E, F khng ph thuc vo v tr ca im M.

b) So snh 2 vc t .Bai 11. Cho t gic ABCD.

a) Hy xc nh v tr ca im G sao cho: (G gl trng tm ca t gic ABCD).

b) Chng minh rng vi im O tu , ta c: .Bai 12. Cho G l trng tm ca t gic ABCD. A(, B(, C(, D( ln lt l trng tm ca cc tam gic BCD, ACD, ABD, ABC. Chng minh:

a) G l im chung ca cc on thng AA(, BB(, CC(, DD(.

b) G cng l trng tm ca ca t gic A(B(C(D(.Bai 13. Cho t gic ABCD. Trong mi trng hp sau y hy xc nh im I v s k sao cho cc vect u bng vi mi im M:

a)

b)

c)

d) .VN 4: Chng minh ba im thng hng Hai im trng nhau

( chng minh ba im A, B, C thng hng ta chng minh ba im tho mn ng thc , vi k ( 0.

( chng minh hai im M, N trng nhau ta chng minh chng tho mn ng thc , vi O l mt im no hoc .Bai 1. Cho bn im O, A, B, C sao cho : . Chng t rng A, B, C thng hng.Bai 2. Cho hnh bnh hnh ABCD. Trn BC ly im H, trn BD ly im K sao cho: . Chng minh: A, K, H thng hng.

HD: . Bai 3. Cho (ABC vi I, J, K ln lt c xc nh bi: , , .

a) Tnh . (HD: )

b) Chng minh ba im I, J, K thng hng (HD: J l trng tm (AIB).

Bai 4. Cho tam gic ABC. Trn cc ng thng BC, AC, AB ln lt ly cc im M, N, P sao cho , , .a) Tnh theo .

b) Chng minh ba im M, N, P thng hng.

Bai 5. Cho hnh bnh hnh ABCD. Trn cc tia AD, AB ln lt ly cc im F, E sao cho AD = AF, AB = AE. Chng minh:a) Ba im F, C, E thng hng.

b) Cc t gic BDCF, DBEC l hnh bnh hnh.Bai 6. Cho (ABC. Hai im I, J c xc nh bi: ,

. Chng minh 3 im I, J, B thng hng.Bai 7. Cho (ABC. Hai im M, N c xc nh bi: , . Chng minh 3 im M, G, N thng hng, vi G l trng tm ca (ABC.Bai 8. Cho (ABC. Ly cc im M N, P:

a) Tnh .b) Chng minh 3 im M, N, P thng hng.Bai 9. Cho (ABC. V pha ngoi tam gic v cc hnh bnh hnh ABIJ, BCPQ, CARS. Chng minh cc tam gic RIP v JQS c cng trng tm.Bai 10. Cho tam gic ABC, A( l im i xng ca A qua B, B( l im i xng ca B qua C, C( l im i xng ca C qua A. Chng minh cc tam gic ABC v A(B(C( c chung trng tm.Bai 11. Cho (ABC. Gi A(, B(, C( l cc im nh bi: , , . Chng minh cc tam gic ABC v A(B(C( c cng trng tm.Bai 12. Trn cc cnh AB, BC, CA ca (ABC ly cc im A(, B(, C( sao cho:

Chng minh cc tam gic ABC v A(B(C( c chung trng tm.Bai 13. Cho tam gic ABC v mt im M tu . Gi A(, B(, C( ln lt l im i xng ca M qua cc trung im K, I, J ca cc cnh BC, CA, AB.

a) Chng minh ba ng thng AA(, BB(, CC( ng qui ti mt im N.

b) Chng minh rng khi M di ng, ng thng MN lun i qua trng tm G ca (ABC.

Bai 14. Cho tam gic ABC c trng tm G. Cc im M, N tho mn: , . Chng minh ng thng MN i qua trng tm G ca (ABC.Bai 15. Cho tam gic ABC. Gi I l trung im ca BC, D v E l hai im sao cho .

a) Chng minh .

b) Tnh . Suy ra ba im A, I, S thng hng.Bai 16. Cho tam gic ABC. Cc im M, N c xc nh bi cc h thc , .

a) Xc nh x A, M, N thng hng.

b) Xc nh x ng thng MN i trung im I ca BC. Tnh .Bai 17. Cho ba im c nh A, B, C v ba s thc a, b, c sao cho .

a) Chng minh rng c mt v ch mt im G tho mn .

b) Gi M, P l hai im di ng sao cho . Chng minh ba im G, M, P thng hng.

Bai 18. Cho tam gic ABC. Cc im M, N tho mn .

a) Tm im I tho mn .

b) Chng minh ng thng MN lun i qua mt im c nh.

Bai 19. Cho tam gic ABC. Cc im M, N tho mn .

a) Tm im I sao cho .

b) Chng minh rng ng thng MN lun i qua mt im c nh.

c) Gi P l trung im ca BN. Chng minh ng thng MP lun i qua mt im c nh.

VN 5: Tp hp im tho mn ng thc vect

tm tp hp im M tho mn mt ng thc vect ta bin i ng thc vect a v cc tp hp im c bn bit. Chng hn:

Tp hp cc im cch u hai u mt ca mt on thng l ng trung trc ca on thng .

Tp hp cc im cch mt im c nh mt khong khng i ng trn c tm l im c nh v bn knh l khong khng i.

Bai 1. Cho 2 im c nh A, B. Tm tp hp cc im M sao cho:

a)

b) .

HD: a) ng trn ng knh ABb) Trung trc ca AB.Bai 2. Cho (ABC. Tm tp hp cc im M sao cho:

a)

b)

c)

d) .

HD: a) Trung trc ca IG (I l trung im ca BC, G l trng tm (ABC).

b) Dng hnh bnh hnh ABCD. Tp hp l ng trn tm D, bn knh BA.Bai 3. Cho (ABC.

a) Xc nh im I sao cho: .

b) Chng minh rng ng thng ni 2 im M, N xc nh bi h thc:

lun i qua mt im c nh.

c) Tm tp hp cc im H sao cho: .

d) Tm tp hp cc im K sao cho:

Bai 4. Cho (ABC.

a) Xc nh im I sao cho: .

b) Xc nh im D sao cho: .

c) Chng minh 3 im A, I, D thng hng.

d) Tm tp hp cc im M sao cho: .

1. Trc to

( Trc to (trc) l mt ng thng trn xc nh mt im gc O v mt vect n v . K hiu .

( To ca vect trn trc:

.

( To ca im trn trc:

.

( di i s ca vect trn trc:

.

Ch :+ Nu th .

Nu th .

+ Nu A(a), B(b) th .

+ H thc Sal: Vi A, B, C tu trn trc, ta c: .2. H trc to

( H gm hai trc to Ox, Oy vung gc vi nhau. Vect n v trn Ox, Oy ln lt l . O l gc to , Ox l trc honh, Oy l trc tung.

( To ca vect i vi h trc to :

.

( To ca im i vi h trc to :

.

( Tnh cht: Cho , :

+

+

+

+ cng phng vi ( (k ( R: .

( (nu x ( 0, y ( 0).

+ .

+ To trung im I ca on thng AB: .

+ To trng tm G ca tam gic ABC: .

+ To im M chia on AB theo t s k ( 1: .

( M chia on AB theo t s k ( ).VN 1: To trn trcBai 1. Trn trc x'Ox cho 2 im A, B c ta ln lt l (2 v 5.

a) Tm ta ca .

b) Tm ta trung im I ca on thng AB.

c) Tm ta ca im M sao cho .

d) Tm ta im N sao cho .

Bai 2. Trn trc x'Ox cho 2 im A, B c ta ln lt l (3 v 1.

a) Tm ta im M sao cho .

b) Tm ta im N sao cho .

Bai 3. Trn trc x'Ox cho 4 im A((2), B(4), C(1), D(6).

a) Chng minh rng: .

b) Gi I l trung im ca AB. Chng minh: .

c) Gi J l trung im ca CD. Chng minh: .

Bai 4. Trn trc x'Ox cho 3 im A, B, C c ta ln lt l a, b, c.

a) Tm ta trung im I ca AB.

b) Tm ta im M sao cho .

c) Tm ta im N sao cho .

Bai 5. Trn trc x'Ox cho 4 im A, B, C, D tu .

a) Chng minh: .

b) Gi I, J, K, L ln lt l trung im ca cc on AC, BD, AB, CD. Chng minh rng cc on IJ v KL c chung trung im.VN 2: To trn h trcBai 1. Vit ta ca cc vect sau:

a) .

b) .

Bai 2. Vit di dng khi bit to ca vect l:

a) .

b) .

Bai 3. Cho . Tm to ca cc vect sau:

a) .b) .

Bai 4. Cho .

a) Tm to ca vect .

b) Tm 2 s m, n sao cho: .

c) Biu din vect .Bai 5. Cho hai im .

a) Tm to im C sao cho: .

b) Tm im D i xng ca A qua C.

c) Tm im M chia on AB theo t s k = 3.Bai 6. Cho ba im A(1; 1), B(1; 3), C(2; 0).

a) Chng minh ba im A, B, C thng hng.

b) Tm cc t s m im A chia on BC, im B chia on AC, im C chia on AB.

Bai 7. Cho ba im A(1; (2), B(0; 4), C(3; 2).

a) Tm to cc vect .

b) Tm ta trung im I ca on AB.

c) Tm ta im M sao cho: .

d) Tm ta im N sao cho: .

Bai 8. Cho ba im A(1; 2), B(2; 3), C(1; 2).

a) Tm to im D i xng ca A qua C.

b) Tm to im E l nh th t ca hnh bnh hnh c 3 nh l A, B, C.

c) Tm to trng tm G ca tam gic ABC.BI TP N CHNG IBai 1. Cho tam gic ABC vi trc tm H, B( l im i xng vi B qua tm O ca ng trn ngoi tip tam gic. Hy xt quan h gia cc vect .Bai 2. Cho bn im A, B, C, D. Gi I, J ln lt l trung im ca AB v CD.

a) Chng minh: .

b) Gi G l trung im ca IJ. Chng minh: .

c) Gi P, Q l trung im ca cc on thng AC v BD; M, N l trung im ca cc on thng AD v BC. Chng minh rng ba on thng IJ, PQ v MN c chung trung im.

Bai 3. Cho tam gic ABC v mt im M tu .

a) Hy xc nh cc im D, E, F sao cho , , . Chng minh cc im D, E, F khng ph thuc vo v tr ca im M.

b) So snh hai tng vect: v .

Bai 4. Cho (ABC vi trung tuyn AM. Gi I l trung im AM.

a) Chng minh: .

b) Vi im O bt k, chng minh: .

Bai 5. Cho hnh bnh hnh ABCD tm O. Gi I l trung im BC v G l trng tm (ABC. Chng minh:

a) .

b) .

Bai 6. Cho hnh bnh hnh ABCD tm O. Gi I v J l trung im ca BC, CD.

a) Chng minh:

b) Chng minh: .

c) Tm im M tho mn: .

Bai 7. Cho tam gic ABC c trng tm G. Gi D v E l cc im xc nh bi , .

a) Tnh .

b) Chng minh ba im D, E, G thng hng.Bai 8. Cho (ABC. Gi D l im xc nh bi v M l trung im on BD.

a) Tnh theo .

b) AM ct BC ti I. Tnh v .

Bai 9. Cho (ABC. Tm tp hp cc im M tha iu kin:

a)

b)

c)

d)

e)

Bai 10. Cho (ABC c A(4; 3) , B((1; 2) , C(3; (2).

a) Tm ta trng tm G ca (ABC.

b) Tm ta im D sao cho t gic ABCD l hnh bnh hnh.Bai 11. Cho A(2; 3), B((1; (1), C(6; 0).

a) Chng minh ba im A, B, C khng thng hng.

b) Tm ta trng tm G ca (ABC.

c) Tm ta im D t gic ABCD l hnh bnh hnh.Bai 12. Cho A(0; 2) , B(6; 4) , C(1; (1). Tm to cc im M, N, P sao cho:

a) Tam gic ABC nhn cc im M, N, P lm trung im ca cc cnh.

b) Tam gic MNP nhn cc im A, B, C lm trung im ca cc cnh.

I. VECT

CHNG I

VECT

II. TO

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