phần loại và phương pháp giải hình học 10

Dinh nghia vecto: Vccta la doan thang co huong, nghia la trong hai diem m u t cua doan thang da chi ro diem nao la diem dau, diem nao la diem cuoi.. Co bao nhieu vecto khac vecto-khong

516.0076

PH121L

) A U THANH KY - NGUY§N M J N H NHIEN "

_JYEN ANH TRUING - NGUYEN TAN SIENG

D6 N G O G T H U Y (Nhom giao vien chuyen toan trifdng THPT)

PHANLOAI

mmm WHIM

(Tai Ban, SCfa ChOa Va Bo Sung)

• Danh cho hoc sinh Idp 10 on tap va nang cao kien thufc

• Bien soan theo noi dung sach giao khoa cua bp GD&DT

AT BAN DAI HOC QUOC GIA HA NOI

1

w a\ DAU THANH KY - NGUYEN MINH NHIEN 7«3UYElsrPFrtrKFwi\iFr

NGUYEN ANH TRUdNG - NGUYEN TAN SIENG

D6 N G O C T H U Y (Nhom glao vien chuyen toan trudng THPT)

if I ! ri; :

PEIANLQAI

& FHlTdNG PHAP GIAI

(Tai Ban, Sufa ChOa Va Bo Sung)

Danh cho hoc sinh Idp 10 on tap va nang cao kien thiJc Bien soan theo npi dung sach giao khoa cua bp GD&DT

mm, ^ ^ ^ - - ^ A l l j i/Sn riiA HA NOI

Cac em hoc sinh than men!

"Phan loai va phuong phap giai Hinh hoc 10" la mot trong nhi>ng cuon

thuoc bo sach "Phan loai va phuang phap giai lop 10,11,12 ", do nhom tac gia

chuyen toan T H P T bien soan

Vai each vie't khoa hoc va sinh dpng, cuon sach se g i u p ban doc tiep can voi

mon Toan mot each t u nhien, khong ap luc, ban doc trd nen t u tin va nang

dong hon; hieu ro ban chat, biet each phan tich de t i m ra trong tarn cua van de

va biet giai thich, lap luan cho tung bai toan Su da dang cua he thong bai tap

va tinh huong giup ban doc iuon hung thu khi giai toan

Tac gia chii trong bien soan nhung cau hoi ma, noi d u n g ca ban bam sat

sach giao khoa va cau true de thi Dai hoe, dong thai phan bai tap thanh cac

dang toan eo 161 giai chi tie't Hien nay de thi Dai hoc khong kho, to hgp cua

nhieu van de d o n gian, nhung chiVa nhieu cau hoi m a neu khong nam chac ly

thuyet se lung tung trong viec tim loi giai bai toan Vai mot bai toan, khong

nen thoa man ngay voi mot loi giai minh vua tim dugc ma phai c o g a n g t i m

nhieu each giai nhat cho bai toan dcS, moi mot each giai se c6 them phan kien

thue moi on tap

M o n Toan la mot mon rat ua phong each tai tu, n h u n g phai la tai t u mot

each sang tao va thong m i n h K h i giai mot bai toan, thay v i dCing thoi gian de

luc loi tri nha, thi ta can phai suy nghi phan tich de t i m ra p h u a n g phap giai

quyet bai toan do Doi voi Toan hoc, khong c6 trang sach nao la thua T u n g

trang, t u n g dong deu phai hieu Mon Toan doi hoi phai kien nhan va ben bi

ngay tu n h u n g bai tap dan gian nhat, nhi>ng kien thue co ban nhat Vi chinh

nhiing kien thue co ban m o i giup ban doe hieu dugc n h u n g kien thue nang cao

sau nay

L u d w i g Van Beethoven t u n g noi: "Gigt nuoe co the lam mon tang da,

khong phai v i gigt nuoe co sue manh, ma do nuae chay lien tuc ngay dem Chi

CO sir phaii da'u khong met moi moi dem lai tai nang Do do ta co the khSng

djnh, khong nhich tung buae thi khong bao gio co the di xa ngan d a m "

Mac d i i tac gia da danh nhieu tam huyet cho cuon sach, song su sai sot la

dieu kho tranh khoi Chung toi rat mong nhan dugc su phan bien va gop y quy

bau ciia quy doe gia de nhung Ian tai ban sau cuon sach dugc hoan thi^n han

*' , r\ \ ^V'Q' ^^^y "^^^ nhom bien soan

y\n Phu Khanh

1. Dinh nghia vecto:

Vccta la doan thang co huong, nghia la trong hai diem m u t cua doan thang da chi ro diem nao la diem dau, diem nao la diem cuoi

Vecto CO diem dau la A, diem cuoi la B ta k i hieu: AB

Vecto eon dugc k i hieu la: a, b, x, y,

Veeta - khong la vecto co diem dau trung diem cuoi K i hieu la 0

2 Hai vecto cung phuomg, citng huong

- D u o n g thang d i qua diem dau va diem cuoi ciia vecto ggi la gid cua vecto

- H a i vecto co gia song song hoac triing nhau ggi la hai vecto ciing phwang

- H a i vecto ciing p h u a n g thi hoac eung huong hoac nguge h u o n g

c D Hinh 1.2 H

Vi du: 6 hinh ve tren tren (hinh 2) thi hai vecto A B va C D ciing h u o n g eon

EF va H G nguge huong

Dac biet: vecto - khong cung huong voi mgi vec to

3 Hai vecto bSng nhau A B

- D g dai doan thSng AB ggi la do dai vecto

A B , k i h i e u AB Vgy AB = A B

B CAC D A N G T O A N VA P H l T a N G PHAP G I A I

DANG TOAN 1: XAC DINH MOT VECTO; PHUONG, HUONG CUA

' VECTO; DQ DAI CUA VECTO

Vi du 1: Cho tu giac ABCD Co bao nhieu vecto khac vecto-khong c6 diem

dau va diem cuoi la dinh cua tu giac

Lai gidi

Hai diem phan biet, chang han A, B ta xac djnh dupe hai vecto khac

vecto-khong la AB, BA Ma tu bon dinh A, B, C, D ciia tu giac ta c6 6 cap diem

phan biet do do c6 12 vecto thoa man yeu cau bai toan

Vi du 2: Chung minh rang ba diem A,B,C phan bi^t thang hang khi va chi

khi A B , A C ciing phuong

Lcri giai

Neu A , B , C thang hang suy ra gia cua AB, AC deu la dudng thang di qua

ba diem A,B,C nen AB, AC cung phuong i

Ngup-c lai neu AB, AC ciing phuong khi do duong thSng AB va AC song

song hoac triing nhau Nhung hai duong thang nay ciing di qua diem A nen

hai duong thang AB va AC triing nhau hay ba diem A, B, C thing hang

Vi du 3: Cho tam giac ABC Goi M,N,P Ian lupt la trung diem cua

BC,CA,AB

a) Xac djnh cac vecto khac vecto - khong ciing phuong v6i M N c6 diem dau

va diem cuoi lay trong diem da cho

b) Xac dinh cac vecto khac vecto - khong ciing huong vdi AB c6 diem dau

va diem cuoi lay trong diem da cho

c) Ve cac vecto bang vecto NP ma c6 diem dau A, B

Lcri gidi (Hinh 1.4)

a) Cac vecto khac vecto khong ciing phuong voi M N la

N M , AB, BA, AP, PA, BP, PB

b) Cac vecto khac vecto - khong cung huong voi AB la AP, PB, N M c) Tren tia CB lay diem B' sao cho BB' = NP ^ ' ^

Khi do ta CO BB' la vecto c6 diem

dau la B va bang vecto NP '

Qua A dung duong thang song

song voi duong thang NP

Tren duomg thang do lay diem

A ' sao cho A A " ciing huong

vo-iNP va A A " = NP Hinh 1 1 "

Khi do ta CO A A ' la vecto c6 diem dau la A va bang vecto NP

Vi du 4: Cho hinh vuong ABCD tam O canh a Goi M la trung diem cua

AB, N la diem doi xung voi C qua D Hay tinh do dai ciia vecto sau:

M D , M N ^

Lai gidi (hinh 1.5)

Ap dung djnh ly Pitago trong tam giac vuong M A D ta c6 DM^ = A M ^ + AD^ =

Suy ra MD = M D

-a' + a = — 2 5a'

Qua N ke duong thing song song vol AD cat AB tai P | Khi do tu giac ADNP la hinh vuong va|

3 B A I T A P L U Y £ N T A P * ^ " " ^

Bai 1.1: Cho ngii giac ABCDE Co bao nhieu vecto khac vecto-khong c6 diem

dau va diem cuoi la dinh ciia ngii giac ^

Huang dan gidi

I Hai diem phan biet, chang han A, B ta xac dinh dupe hai vecto khac khong la AB, BA Ma tir nam dinh A, B, C, D, E ciia ngu giac ta c6 10 ca

vecto-I diem phan biet do do c6 20 vecto thoa man yeu cau bai toan

P

Bai 1.2: Cho ba diem A, B, C phan bi^t thang hang

a) Khi nao thl hai vecto AB va AC cung huong ? ^

b) Khi nao thi hai vecto AB va AC ngiroc huong ?

Huang dan gidi ^

a) A nam ngoai doan BC

b) A nam trong doan BC

Bai 1.3: Cho bon diem A, B, C, D phan biet

a) Neu AB = BC thi c6 nhan xet gi ve ba diem A, B, C f>r{>

b) Neu AB = DC thi co nhan xet gi ve bon diem A, B, C, D

Huang dan gidi , (^fy.-j

a) B la trung diem cua AC

b) A, B, C, D thang hang hoac ABCD la hinh binh hanh, hinh thoi, hinh vuong,

hinh chij' nhat

Bai 1.4: Cho hinh thoi ABCD c6 tam O Hay cho biet khang djnh nao sau day

dung ?

^ b) AB = DC c) OA = -OC

a) AB = BC

d) OB = OA e) AB BC f) 2 OA BD

Huang dan gidi

a) Sai b) Dung c) Dung

d) Sai e) Dung f) Sai

Bai 1.5: Cho luc giac deu ABCDEF tam O Hay tim cac vecto khac vecto-khong

CO diem dau, diem cuoi la dinh ciia luc giac va tam O sao cho

a) Bang voi A B b) Ngugc huong voi OC

Huang dan gidi

a) Fd,OC,ED b) c6,OF,BA,DE

Bai 1.6: Cho hinh vuong ABCD canh a , tam O va M la trung diem AB

Tinh dp dai cua cac vecto AB, AC, OA, OM, OA + OB

Huang dan gidi

-Ggi E la diem sao cho tu giac OBEA la hinh

binh hanh khi do no cung la hinh vuong

Ta CO OA + OB = OE => OA + OB = OE = AB = a

Hinh 1.40

Bai 1.7: Cho tam giac ABC deu canh a va G la trong tam Goi I la trung

diem cua A G Tinh do dai cua cac vecto AB, AG, B I

Ta C O AB = AB = a Goi M la trung diem cua BC

Ta C O

AG = AG = - A M = - 7 A B 2 - B M 2 = ^ - A M = -\// 3 3

BI = BI = N/BM'+MI2 = ^ ^ + ^ = — — j _ ( a W ^ a V 2 T

Bai 1.8: Cho truoc hai diem A,B phan biet Tim tap hop cac diem M thoa man

Huang dan gidi

o M A = MB => Tap hop diem M la duong trung true cua doan

M A MB

MA MB thang AB

D A N G T O A N 2: C H U N G M I N H H A I V E C T O BANG N H A U :

1 PHLTONG P H A P G l A l '

• De chiing minh hai vecto bang nhau ta chung minh chung c6 ciing do dai

va cung huong hoac dua vao nhan xet neu tii giac ABCD la hinh binh hanh

thi AB = DC va AD = BC

2 C A C VI D U y ;,rt 7 ,

V i d « l : C h ^ giac ABCD Goi M, N, P, Q Ian lugt la trung diem AB, BC,

CD, DA Chung minh rSng MN=QP •

' • Lai gidi (hinh 1.6)

Do M , N Ian luat la trung diem cua AB va BC nen M N la duong trung binh cua tam giac ABC Suy ra M N / / A C va M N = i A C (1)

Tuong tu QP la duong trung binh cua tam giac

ADC suy ra QP//AC va QP = ^ A C (2)

Tir (1) va (2) suy ra MN//QP va M N = QP

do do tur giac M N P Q la hinh binh hanh

V a y t a c 6 M N = Q P

Vi du 2: Cho tarn giac A B C c6 trong tarn G Goi I la t r u n g diem ciia BC

D u n g diem B' saocho B'B = A G ^ ,>^,.,i i , ,

a) C h u n g m i n h rang BI = IC % -; • '''-^

b) Goi J la trung diem ciia BB' C h u n g m i n h rang BJ = I G

^ L o i ^ a x (hinh 1.7) a) V i I la t r u n g diem cua BC nen BI = C I va BI ciing Huang v o i I C do do

hai vecto B I , IC bang nhau hay BI = IC

b) T a c o B'B = A G suy ra B"B = A G va B B 7 / A G

D o do BJ, I G cung huong (1)

V i G la trong tam tam giac ABC nen

* I G = ^ A G , J la trung diem BB' suy ra

BJ

=1BB-^ V i v a y BJ = I G (2)

I T u (1) va (2) ta CO BJ = I G

Hinh 1.7

Vi du 3: Cho hinh binh hanh A B C D Tren cac doan thSng DC, A B theo t h u

hr lay cac diem M , N sao cho D M = B N Goi P la giao diem ciia A M , DB

va Q la giao diem cua C N , D B C h u n g m i n h rang A M = N C va

D P = Q B

LOT ^ « » (hinh 1.8)

Ta CO D M = B N => A N = M C , mat khac A N song song v o i M C

do do t u giac A N C M la hinh binh hanh

M a t khac D P M = APB (do! dinh)

va A P Q = N Q B (hai goc dong vj) suy ra D M P = B N Q

D o d o A D M P = ABNQ (c.g.c) suy ra DP = QB

De thay DP, QB ciing h u o n g v i vay D P = Q B

8

3 B A I T A P L U Y C N T A P Bai 1.9: Cho t u giac A B C D Goi M , N , P, Q Ian lugt la trung diem AB, BC, CD,

D A C h u n g m i n h rang M Q = NP m-ofi

Huong dan gidi • V ^ * * ^

D o M , Q Ian lugft la trung diem ciia AB va A D nen A

M Q la d u o n g trung binh ciia tam giac A B D suy

A B ; P la giao diem ciia A M , DB va Q la giao diem ciia C N , DB C h u n g

m i n h rang D M = NB va DP PQ = QB

Huang dan gidi '

Ta CO t u giac D M B N la hinh binh hanh v i D M = NB = | A B , D M / /NB

Suy ra D M = N B Xet tam giac C D Q c6 M la trung diem ciia

DC va M P / / Q C do do P la trung diem ciia

D Q T u o n g t u xet tam giac ABP suy ra dugc Q la trung diem ciia PB

V i vay DP = PQ = QB t u do suy ra DP = PQ = QB Bai 1.11: Cho hinh thang A B C D c6 hai day la A B va C D v o i A B - 2 C D T u C

ve C i = D A C h u n g m i n h rang

Huang dan gidi

a) A D = IC va D I = CB a) Ta CO C I = D A suy ra A I C D la hinh binh hanh

= > A D = IC

Ta CO D C = A I ma A B - 2CD do do A I = i A B

=> I la trung diem A B

Ta CO DC = IB va DC / /IB => tir giac BCD! la hinh binh hanh

Suy ra DI = CB

b) I la trung diem cua AB=>AI = IB va tu giac BCDI la hinh binh hanh

=> IB = DC suy ra A I = IB = DC

Bai 1.12: Cho tam giac ABC c6 true tarn H va O tarn la duong tron ngoai tiep

Goi B' la diem doi xung B qua O Chung minh: A H =B'C

Huang dan gidi

a) Dinh nghla: Cho hai vecto a; b Tir diem A tuy y ve AB = a roi tu B ve

BC = b khi do vecto AC duoc goi la tong cua hai vecto a; b

Ki hieu AC = a + b (Hinh 1.9)

b) Tinh chat:

+ Giac hoan : a + b = b + a

+ Ket hop : (a + b) + c = a + (b + c)

+ Tinh chat vecto - khong: a + 0 = a, Va

2 Hieu hai vecta Hinh 1.9 ?

a) Vecto doi cua mot vecto

Vecta doi ciia vecto a la vecto ngugc huang va cung do dai vol vecto a

Ki hieu -a

Nhu vay a + (-a) = 0, Va va AB = -BA

b) Djnh nghta hieu hai vecto:

Hieu ciia hai vecta a va b la tong ciia vecto a va vecto doi ciia vecto b Ki

hi^u la a - b = a + (-b)

3 Cac quy tic:

Quy tac ba diem: Cho A, B ,C tuy y, ta c6 : AB + BC = AC

Quy tic hinh binh hanh: Neu ABCD la hinh binh hanh thi AB + AD = AC

Quy tac ve hieu vecto: Cho O, A, B tiiy y ta c6: OB - O A = AB

10

Chu ij: Ta c6 the mo rpng quy tic ba diem cho n d i e m A , , A 2 , , A n thi

A1A2 + A2A3" + + A„_,An = A, A„

B C A C D A N G T O A N V A P H l / Q N G P H A P G I A I

D A N G T O A N 1: XAC DINH DO DAI TONG, HIEU CUA CAC VECTO

1 PHLTONG P H A P G I A I

De xac dinh do dai tong hieu ciia cac vecto

• Truoc tien sir dung djnh nghla ve tong, hieu hai vecto va cac tinh chat, quy

tac de xac djnh phep toan vecto do

• Dua vao h'nh chat cua hinh, sir dung djnh li Pitago, he thuc lugng trong tam

giac vuong de xac djnh do dai vecto do

2 C A C V i D U

Vidu 1: Cho tam giac ABC vuong tai A c6 ABC = 30" va BC = aVS

Tinh do dai ciia cac vecto AB + BC, AC - BC va AB + AC

Lai gidi (hinh 1.10)

Theo quy tac ba diem ta c6 AB + BC = AC

Goi D la diem sao cho tir giac ABDC la hinh binh hanh

Khi do theo quy tic hinh binh hanh ta c6 AB + AC = AD

Vi tam giac ABC vuong 6 A nen tir giac ABDC la hinh chu nhat suy ra

A D = BC = a>/5 Vay AB + AC AD = A D = a>/5

11

Vtdu 2: Cho hinh vuong ABCD c6 tarn la O va canh a M la mpt diem bat ky

a) Tinh AB + A D OA - BO CD - DA

b) Chung minh rang u = M A + MB - MC - M D khong phu thuoc vi tri diem

M Tinh do dai vecto u

Lai gidi {hinh 1.11)

a) + Theo quy tac hinh binh hanh ta c6 AB + AD = AC

Suy ra A B + A D AC = A C

Ap dung dinh li Pitago ta c6

AC^ = AB^ + BC^ = 2a^ ^ AC = >/2a

h Ma |BD| = B D = 7AB^ + AD^ = suy ra |CD - D A | = a^f2

b) Theo quy tSc phep tru ta c6: u = (MA - MC) + (MB - M D ) = C A + DB

Suy ra u khong phu thuoc vj tri diem M

Qua A ke duong thang song song v6i DB cat BC tai C

Khi do tu giac ADBC la hinh binh hanh (vi co cap canh doi song song) suy

^ Huong dan gidi '^A'

Theo quy tac phep tru ta c6

A B - A C = CB: A B - A C =BC = a

Gpi A' la dinh ciia hinh binh hanh A B A C va

O la tam hinh binh hanh do Hinh 1.45

b) Tinh do dai vecto M A - MB - MC + MD

Huang dan gidi

Huang dan gidi

a) Tu gia thie't suy ra ba diem A, B, C tao thanh tam giac deu nhan O lam trongtamdodo AOB = BOC=COA = 120" ,

b) Gpi I la trung diem BC Theo cau a) AABC deu nen AI = — a

2

O B + A C - O A " =a>/3

Bai 1.17: Cho goc Oxy Tren Ox, Oy lay hai diem A, B Tim dieu ki?n cua A,B

sao cho OA + OB nam tren phan giac ciia goc Oxy !

Huang dan gidi

Dung hinh binh hanh OACB Khi do: OA+OB = OC

Vay OC nam tren phan giac goc xOy <=> OACB la hinh thoi <=> OA = OB

D A N G T O A N 2: CHUNG MINHDANG THUC VECTO

1 PHLTONG PHAPGIAI

• De chung minh dang thuc vecta ta c6 cac each bien doi: venay thanh vekia,

bien doi tuong duong, bien doi hai veciing bang mot dai lirong trung gian

Trong qua trinh bien doi ta can su dung linh boat ba quy t3c ti'nh vecto

Lmi I/: Khi bien doi can phai liitmi<^ dicli, chang han bien doi vc'phai, ta can

xem vetrai c6 dai luong nao de tu do lien tuang den kie'n thuc da c6 de lam

sao xuat hien cac dai lugng 6 vetrai Va ta thuang bien doi ve'phuc tap ve

vedan gian hon

2 CAC Vi DU

Vtdu 1: Cho nam diem A,B,C,D,E Chung minh r5ng

a M B +CD + EA = CB + ED b) AC + C D - E C = A E - D B + CB

Lai gidi

a) Bien doi vetrai ta c6 ^

VT = (AC + CB) + C b + (Eb + DA)

= (CB + Eb) + (AC + Cb) + DA =(CB + Eb) + AD + DA

= CB + ED = VP (DPCM) :

b) Dang thuc tuong duong voi

( A C - A E ) + ( C D - C B ) - E C + DB = d<:>EC + B D - E C + DB = 0

BP + DB = 6 (dung) (DPCM)

Vidu 2: Cho hinh binh hanh ABCD tam O M la mot diem bat ki trong m|it

phang Chung minh rang

Ac;

M A - M B = M D - M C » B A - C D (dung do A B C D la hinh binh hanh)

Vi du 3: Cho tam giac ABC Goi M, N, P Ian lugt la trung diem cua

BC, CA, AB Chung minh rang a) BM + CN +AP = d

b) AP + A N - A C + BM = d c) OA + OB + OC = OM + OIV + OP voi O la diem bat ki '

Lai gidi {H\nh 1.13)

a) Vi P N , M N la duang trung binh cua tam giac ABC nen PN//BM, M N / / B P suy ra tu giac BMNP la hinh binh hanh

BM = PN

N la trung diem cua A C => C N = NA

Do do theo quy tac ba diem ta co

c) Theo quy tac ba diem ta c6

„ ' • Huang dan gidi

a) Ap dung quy tac phep tru ta c6 _

D A - C A = D B - C B < » D A - D B = C A - d B c>BA = BA (dung) V

b) A p dung quy tac ba diem ta c6

A C + DA + BD = A D - CD + BA » (DA + AC) + BD = (BA + A D ) - C D

o DC + BD = 150 - CD (diing)

Bai 1.19: Cho cac diem A , B, C , D , E , F

Chung minh rang A D + BE + CF = AE + BF + CD

Hu&ng dan gidi Cdch 1: Dang thuc can chung minh tuang duong voi

Bai 1.20: Cho hinh binh hanh ABCD tam O M la mot diem bat ki trong mat

phang Chung minh r^ng

Ma ob = BO suy ra BA + BC + OB = MO - MB

Bai 1.21: Cho tam giac ABC Goi M, N , P Ian lugt la trung diem ciia

BC, CA, AB Chung minh rang a) NA + PB + MC = 0 b) MC + BP + NC = BC

Huang dan gidi ^

Huang dan gidi ,

Theo quy tac trir va quy tac hinh binh hanh ta CO ^ , ,.<^

B ' B + CC' + ITD = ( A B - AB') + (AC' - AC) + (AD - A D '

j^pl = ( A B + A D ) - A C - ( A B ' + A D ' ) + A C = 0

Chung minh rang OA + OB + OC + OE + OF = 0

Huang dan gidi

Dat u = OA + OB + OC + OE + OF *

Vi ngu giac deu nen vecto OA + OB + OC + OE cung phuong voi OF nen

u cung phuong voi OF • / <

Tuong t u u cung phuong voi OE suy ra u = 0 ^ •

Bai 1.24: Cho hinh binh hanh ABCD Dung A M = BA, M N = DA, NP = DC,

PQ = BC •••

Chung minh rang: AQ = 0 A;

' Theo quy tac ba diem ta c6 AQ = A M + MN + NP + PQ = BA + DA + DC + BC

Mat khac BA + BC = BD, DA + DC = DB suy ra AQ = BD + DB = 0

§ 3 TiCH CUA MOT VECTO VOI MOT SO

A TOM T A T L Y T H U Y E T

1 Dinh nghia: Tich ciia vecta a vai so thuc k ^ 0 la mpt vecto, ki hieu la ka

C l i n g huong voi a neu k > 0, ngUQic huong vai a neu k < 0 va c6 dp dai

bang |k| a •

Quy uoc: Oa = 0 va kO = 0

2 Tinhchat:

i) (k + m)a = ka + ma ii) k(a ± b) = ka ± kb

iii) k(ma) = (km)a iv) ka = Oc:> _ ^ ^ ,

v) la = a, (-l)a = -a

3 Dieu k i f n de hai vecta cung phirong

• b cung phuong a (a 5^ 0) khi va chi khi CO so k thoa b = ka

• Dieu ki^n can va dii de A, B, C thang hang la c6 so' k sao cho AB = kAC

4 Phan tich mpt vecto theo hai vecto khong cung phuong

Cho a khong cung phuong b Voi moi vecta x luon duoc bieu dien

X = ma + nb vai m, n la cac so thuc duy nhat

B C A C D A N G T O A N V A P H l / Q N G P H A P G I A I

DANG TOAN 1: DUNG VA TINHDO DAI VECTO CHUA TICH

MOT VECTO VOI MOT SO

1 PHUONG PHAPGlAl

Su dung djnh nghia tich cua mpt vecta vai mpt so' va cac quy tac ve phep toan

• vecta de dung vecto chua tich mpt vecto vai mpt s6^ ke't hap vai cac djnh li

pitago va he thuc lupng trong tam giac vuong de tinh dp dai ciia chiing

2 CAC Vi DU

Vi du 1: Cho tam giac deu ABC canh a diem M la trung diem BC Dung

cac vecta sau va tinh dp dai ciia chung

Khi do ta CO i AB = A N , 2AC = AQ suy ra theo quy tac hinh binh hanh ta

CO i AB + 2AC = A N + AQ = AP Gpi L la hinh chie'u ciia A len PN

8

\idu 2: Cho hinh vuong ABCD canh a , ' /

a) Chung minh r^ng u = 4MA - 3MB + MC - 2Ml5 khong phu thuoc vao vj

tri diem M

b) Tinh do dai vecto u

Lm^»d« (Hinh 1.15) a) Goi O la tarn hinh vuong

Theo quy tac ba diem ta c6

u = 4 ( M d + OA) - 3 ( M d + OB) + (MO + OC) - 2 { M d + O D )

- 4 0 A - 3 0 B + O C - 2 0 D

Ma OD = - O B , OC = - O A nen u = 30A - OB

Suy ra u khong phu thuoc vao vj tri diem M

b) Lay diem A" trcn tia OA sao cho OA' = 30A khi do

OA' = 30A do do u = OA' - OB = BA *

Mat khac BA' = VOB^ +OA'^ = VoB^ +90A^ =aS

Suy ra |u| = a>/5 ' " ^ ^ ^ ^ " '

3 BAI TAP LUY^N TAP

Bai 1.25 Cho tam giac deu ABC canh a Goi diem M , N Ian luot la trung

diem BC, CA Dung cac vecto sau va tinh do dai ciia chung

A B E F , theo quy tSc hinh binh hanh ta

a) Chung minh r3ng u = MA - 2MB + 3MC - 2MD khong phu thuoc vao vj tri

diem M b) Tinh do dai vecto u

Huong dan gidi |

Goi O la tam hinh vuong

Theo quy tac ba diem ta c6

> '* 21

li = ( M O + OA") - 2 ( M O + O B ) + 3{M6 + O C ) - 2 ( M O + O D )

= O A - 2 0 B + 3 0 C - 2 0 D

Ma OD = -OB, OC = -OA nen u = -20A

k Suy ra u khong phu thu(

b) l i = -20A =20A = aN/2

Suy ra u khong phu thugc vao vj tri diem M 1

D A N G T O A N 2: CHUNG MINHDANG THUC VECTO

1 PHUONG PHAPGIAI •

Si'e dun^ cdc kie'n Ihiec sau debich ddi venay thanh vckia hoqc cd hai bieu thirc a

hai vecung blin<f bieu thiec thu ba hoqc bicn dot tuan^ duang vcdang thi'tc dung:

• Cac tinh chat phep toan vecta

• Cac quy t3c: quy tac ba diem, quy tac hinh binh hanh va quy tac phep tru \

• Tinh chat trung diem: j

M la trung diem doan thang AB o MA + MB = 0

M la trung diem doan th5ng AB o OA + OB = 20M (Vai O la diem tuy y)

• Tinh chat trong tam:

G la trong tarn ciia tam giac ABC <=> GA + GB + GC = O

G la trong tam aia tam giac ABC <=> OA +dB+OC = 30G (Voi O la diem tuy y)

2 CAC VI Dg

Vidu 1: Cho tu giac ABCD Goi I, J Ian lugt la trung diem ciia AB va CD, O

la trung diem ciia IJ Chung minh rang:

Vay AC + BD = (Ai + Bl) + ( JC + fd) + 2IJ = 2lj dpcm

b) Theo he thuc trung diem ta c6 OA + OB = 20i, OC + OD = 20]

Mat khac O la trung diem IJ nen OI + OJ = 0 Suyra OA+ OB+ OC + O D = 2(Oi + Oj) = 6 dpcm

c) Theo cau b ta c6 OA + OB + OC + OD = 0 do do vai moi diem M thi

OA + OB + OC + O b = d

o (OM + M A ) + (OM + M B ) + (OM + MC) + ( O M + M D ) = 0 ' ' '

<j> MA + MB + MC + MP = 4MO dpcm

Vi du 2: Cho hai tam giac ABC va AiB^Ci c6 ciing trong tam G Goi

Gi, G2, G3 Ian lugt la trong tam tam giac BCA^, ABCj, ACBj Chung minh rang GG^ + GG2 + GG3 = 0 •

Lai gidi

Vi Gi la trong tam tam giac BCAj nen 3GG,' = GB + GC + GAj

Tuong tu G2, G3 Ian lugt la trong tam tam giac ABC|, ACBj suy ra'

VI du 3: Cho tam giac ABC c6 true tam H, trong tam G va tam duang tron

ngoai tiep O Chung minh rSng: _ _ a)HA + HB + HC = 2Hd b) OA + dB + OC = OH c) GH + 2GO = 0 —

Lmgifli (Hinh 1.17) a) Pe thay HA + HB + HC = 2Hd neu tam giac ABC vuong Neu tam giac ABC khong vuong goi P

la diem doi xung ciia A qua O khi do j BH//PC (vi cung vuong goc vai AC)

BP//CH (vi cung vuong goc voi AB) ' Suy ra BPCH la hinh binh hanh,

do do theo quy tic hinh binh hanh thi ^

HB + HC = HP (1) Mat khac vi O la trung diem ciia AD <

nenHA + HD = 2 H d (2)

Tu (1) va (2) suy ra H A + HB + H C = 2H6 M i n h 1.17

23

b) Theo cau a) ta c6

H A + H B + H C = 2 H O i ,

o ( H O + OA) + ( H O + O B ) + ( H O + Oc) = 2HO

<=> OA + OB + OC = OH dpcm

c) Vi G la trong tam tam giac ABC nen OA + OB + OC = 30G

, , Mat khac theo cau b) ta c6 OA + OB + O C = OH

- Suy ra O H = 30G c:> (OG + G H ) - 30G = 0 « G H + 2Gd = 0

Vi du 4: Cho tam giac ABC voi AB = c, BC = a, CA = b va c6 trong tam G

Goi D,E,F Ian lugt la hinh chieii G len canh BC,CA, AB ^ rlftif i ;|

Chung minh rSng a^ GD + b^ GE + c^ GF = 0 ^

% LmgitJi (hinh 1.18)

Tren tia G D , G E , M P Ian luot lay cac diem N , P, Q sao cho

G N = a, G P = b, G Q = c va dung hinh binh hanh G P R N

Ta CO a ^ G D + b ^ G E + c^GF = 0

<=> a G D G N + b.GE.GP + c.GF.GQ = 0 (*) >

Ta CO a.GD = 2S\(3gc, b.GE = 2S , C.GF = 2S^GAB' "^9*^ khac G la trong

tam tam giac A B C nen S,,GBC S^^GCA = SACAB suy ra a.GD =b.GE =c.GF

Vay (*) G N + G P + G Q = 0

Ta CO A C = G P = b, PR = B C = a ^

va ACB = GPR (goc c6 cap canh vuong goc voi nhau) fj

Suy ra AACB = AGPR(c.g.c) / ^

=> GR = AB = c va POR = BAC

Ta CO QGP + BAC = ] 8 0 " ^ Q G P + GPR = 180°

=> Q, G, R thang hang do do G la trung diem ciia QR

Theo quy tMc hinh binh hanh va h^ thuc trung diem ta c6 yy

GN + GP + GQ = GR + GQ = d ;„

Vay a^GEJ + b^GE + c^ GF = 0 •

Vi du 5: Cho tam giac ABC voi cac canh AB = c, BC = a, CA = b Goi I la

tam duong tron noi tiep tam giac ABC •»* •

Chung minh rang alA + bIB + cIC = 0

ID - IB = ^(IC - I D ) (b + c ) I D = bIB + ciC (1)

Do I la chan duong phan giac nen ta c6 :

Tu (1) va (2) ta CO dieu phai chung minh _

Cdch 2: (hinh 1.20)Qua C dung duong thang song song vai A l cat BI tai B';song

song voi BI cat AI tai A' d

Ta CO IC = IA"' + IB' (*) Theo djnh ly Talet va tinh chat duong phan giac trong ta c6 :

I B ^ B A = £ = , i B - = - b i B ( l ) IB' CA, b c ^ ^

BC, CA, AB Chung minh rang * a a) A M + BN + CP = d

b) OA + OB + OC = OKi + ON + OP voi O la diem bat ky

Huang dan gidi u i

a) A M + BN + Cr'=

j ( A B A C ) l b) O M + ON + OP =

- ^ ( 0 B + 0 ( : ) + i ( 0 C + 0 A ) + l ( 0 A + 0 B ) ^

= 6 A + O B + O C

Bai 1.28: Cho tam giac ABC Goi H la diem doi xung voi B qua G vai G la trong

tam tam giac Chung minh rang ' - fei™ srfal j

= 1( A B + A C ) + i ( B C + B A ) + 1 ( C A + C B ) = 6

Bai 1.31: Cho n vecto doi mot khac phuong va tong ciia n - 1 vecto ba't ki

trong n vecto tren ciing phuong voi vecto con lai Chung minh rang tong n

vecto cho d tren bang vecto 0

Huang dan gidi

Gia sir n vecto la aj, i = l,2, ,n Dat u =aj + + a„

Vi tong ciia n - 1 vecto bat ki trong n vecto tren ciing phuong voi vecto con

lai do do u Cling phuong voi hai vecto aj, aj nen u = 0

Bai 1.32: Cho tam giac ABC voi cac canh AB = c, BC = a, CA = b Goi I la tam

va D, E, F Ian lugt la tiep diem ciia canh BC, CA, AB ciia duong tron noi

tiep tam giac ABC M , N, P Ian lugt la trung diem ciia BC, CA, AB Chung

minh rang:

a) cot—+ cot — B 2 2 C) IA + f C cot — + cot — 2 2 ) I B + cot— + cot — A B^ 2 2J ic = d

c o t y I M + c o t | l N + cot|lP = d c) (b + c - a)IM + (a + c - b)IN + (a + b - c)IP = 0 d) aAD + bBE + cCF = d

• ' Huang dan gidi

a) Ggi r la ban kinh duong tron ngi tiep AABC ta c6

a = r cot —+ cot— B C 2 2 ; b = r cot—+ cot— 2 2 ; c = r

J

Theo vi du 5 ta c6 alA + bIB + cIC = 0

f B C^ cot—+ cot— IA + cot— + cot — IB + cot— + cot— ( c A^ IB + r A B^

Theo cau a) ta c6 c o t y ( i B + ic) + cot|(lA + ic) + c o t y ( l A + I B ) = 6

=>alb + bIE + cIF = ( 2 p - b - c ) l A + ( 2 p - c - a ) l B + ( 2 p - a - b ) l C

= alA + bIB + cIC => a AD + bBE + cCF = 0

Bai 1.33: Cho tam giac ABC M la diem bat ky n^m trong tam giac

Chung minh rang : SMBC M A + S M C A M B + SMABMC=d

Huang dan gidi

A ' C A ' B

Goi A ' la giao diem A M v6i B C ta c6 M A ' = M B + _ M C (*)

Mat khac: — - ^MK^ - ^MAC

Bai 1.34: Cho da giac loi A, A j - A ^ ( n > 3 ) ; e j j < i < n la vecto dan vi vuong

gocvai AjAj^, (xem A„+| = A , ) va huang ra phi'a ngoai da giac

Chung minh rMng: A,A2e] + A2K^e2 + - + A n A , e „ = 0 (dinh ly con nhim)

Huang dan gidi

Ta chung minh bang quy nap

Vai n = 3 dang thuc tro thanh

a.e, +b.e2 +c.e3 = 0

(diing vi dang thuc nay tuang

duong vai dang thuc a bai 1 1 )

Gia su dung voi n = k - l , k >4

Goi e la vecto don vj vuong goc voi

A,A|^_, va huong ra ngoai tam giac

' Theo gia thiet quy nap ta c6

A,A2e,+A2A3e2+ + A , „ 2 A , _ , e , : 2 + A ^ _ , A , ( - e ) = d ( 1 )

MatkhacxettamgiacA,A|^_,A,^ taco >

A]Ak_,e + A^_,Ake^_; + A ^ A , e ^ = 0 ( 2 )

Tir ( 1 ) va (2) suy ra dieu phai chirng minh

Bai 1.35: Cho da giac loi A] A2 An ( n > 3 ) voi I la tam duong tron tiep xiic cac

canh ciia da giac; goi ej,l < i < n la vec to don vi ciing huong voi vec to

A,

l A j Chung minh rang cos—i-e, + cos-^e, + + cos—^e " = 0

Huang dan gidi Goi Bj, i = 1 , 2 , , n la cac tiep diem duang tron noi tiep vai canh A^Aj^j Xettugiac A,B,IB„ c6 A ^ i ; , ! = A^B,! = 9 0 " va B ; A , I = B7A,I

Suy ra B ^ , = B^IA, Mat khac IB = IB„ do do lA, ± B,B„

Tuong tu ta c6 lA; ± Bj„,Bj,i = 2 , 3 , , n

Xet da giac loi B,B2 Bn theo dinh ly con nhim ta c6

B„ B, e; + B, B2 e2 + + B„ _, B„ e„ = 6

Hiiih 1.54

A • A • A

-o IB|.c-os-y-e, + IB2.c-os-Y-e2+••• + I ^ n ' ^ " ^ " ^ ' - ' n

Ma IB, = IB2 = = IB,, suy ra dpcm

Bai 1.36: Cho tam giac ABC vuong tai A I la trung diem cua duang cao AH

Chiing minh rang : a^ lA + b^ IB + c^ IC = 0

Huang dan gidi

DANG TOAN 3: XkC DINH DIEM M THOA MAN MOT

DANG THUC VECTO CHO TRUOC

1 PHUONG PHAPGIAI

• Ta bie'n doi d5ng thuc vecto vc dang A M = a trong do diem A va a da biet Khi do ton tai duy nha't diem M sao cho A M = a, de dung diem M ta lay A lam goc dung mot vecto b5ng vecto a suy ra diem ngon vecto nay chinh la

diemM • j ^ ^ ^

-• Ta bieh doi ve d3ng thuc vecto da biet ciia trung diem doan thang va trong tam tam giac

2 C A C VJ D g

Vidu 1: Cho hai diem A, B phan biet Xac djnh diem M bie't 2 M A - 3MB = 0

LaigidiiKmh 1.21) T a c o 2 M A - 3 M B = ()

« 2 M A - 3 ( M A + AB) = d ^ ^

o A M = 3AB

M nam tren tia A B va A M = 3AB - i i'

Vidu 2: Cho t u giac A B C D Xac djnh diem M , N , P sao cho

a) 2 M A + M B + M C = 0 b) N A + N B + N C + N D = 0

c) 3PA + PB + PC + P D = 6

LOT^ifli(hlnh 1.22) a) Goi I la t r u n g diem BC suy ra M B + M C = 2 M i

D o d o 2 M A + M B + M C = 0

« 2 M A + 2 M i = d c> M A + M I = 0

Suy ra M la trung diem A I

b) Goi K, H Ian luat la trung diem ciia AB, C D ta c6

N A + N B + N C + N D = d <=> 2 N K + 2 N H = 0

« N K + N H = d <r> N la trung diem cua K H

c) Goi G la trong tam tam giac BCD khi do ta c6

PB + PC + PD = 3PG

Suy ra 3PA + re + PC + P D = d<=>3PA + 3PG = 0 Hinh 1.22

<=:>PA + PG = 6<:z>P la trung diem A G

Vi du 3: Cho truoc hai diem A, B va hai so thuc a , P thoa man a + p ^ 0

C h u n g m i n h rang ton tai duy nha't diem I thoa man a l A + piB = 0

T u do, suy ra vai diem bat ki M thi a M A + PMB = (a + p ) M I

Lai gidi

Ta c6: a l A + pre = d <:> a l A + P(IA + AB) = d

0 ( a + p)IA + pAB = d. « ( a + P)Ai = pAB o A I = BA

a + p

V i A , B CO d j n h nen vecto — ^ B A khong doi, do do ton tai d u v nha't diem

a + p • •'

1 thoa man dieu ki^n

T u d o suy ra a M A + pMB = a ( M I + l A ) + P(Mi + IB)

3 ) I A - 2 r B = 6^ b ) J A - J B - 2 J C = 6 ^

c ) K A + KB + K C = BC d) 2 L A - L B + 3 L C = A B + A C

Huong dan gidi

a) I la diem doi x u n g cua A qua B /•}

r r ^ l A B c ) A k = - A B d ) A L = - B C ^ ^ ,

Bai 1.38: Cho t u giac A B C D T i m diem co dinh I va hang so k de he thuc sau

thoa man v o i m o i M

a ) M A + M B + 2 M C = k M I b) 2 M A + 3MB - M D = k M I c) M A + 2MB + 3 M C - 4 M D = k M I

Huang dan gidi

a) Cho M ^ I ^ i A + i B + 2fc = 0 o l i + IC: = 0

V o l J la t r u n g diem cua AB, suy ra I la trung diem ciia JC , >( x

M A + M B + 2 M C = kMi<=>4MI = k M i = > k = 4 ^ b) k = 4, A i = - ( 3 A B - A D ) ^ c) k = 2, I A = 2AB + 3 A C - 4 A D

Bai 1.39: Cho tam giac A B C va ba so thuc a, p, y khong dong thai bang khong

Chung m i n h rang:

a) N e u a + p + y ^ 0 thi ton tai duy nha't diem M sao cho

a M A + pMB + y M C = d

b) N e u a + P + y = 0 t h i khong ton tai diem N sao cho a N A + PNB + y N C = d

Huang dan gidi

a) V i a + p + y # 0 = ^ ( a + p) + (p + Y) + (Y + a ) * 0

Khong mat tinh tong quat gia su a + P ^ 0 => 3 ! D : a D A + pDB = 6 I

Suy ra a M A + pNS + y M C = 0<::>(a + p ) M D + y M C = 0 '

Do d o ton tai d u y nha't diem M

b) Gia sir ton tai d i e m N va a ^ 0 ' / ^ 7

T a c o a N A + pNB+yNC=6<:>CA = — C B (mau thuan v o i A B C la tam giac)

Bai 1.40: Cho n diem A | , A 2 , , A „ va n scV k , , k 2 , , k„ ma

k, + k2 + ••• + k„ = k 0 , , ,

a) C h u n g m i n h rang c6 d u y nhat diem G sao cho

k , G A , + k j G A j + + k „ G A „ = 0

D i e m G n h u the g o i la taiti ti cir cua Iw diciu A j \^(in vai he so' k, Trong

t r u o n g hop cac he so k; bang nhau(ta c6 the chon cackj deu bang 1 ) thi G

goi la fr('//y taifi ci'in he diem A;

b) C h i i n g m i n h rang neu G la tam ti cu noi o cau a) thi voi diem M bat ky ta c6

- ( k , M A i + k 2 M A 2 + + k„MA„") = O G

Ihiang dan giiii

tA v:;,m ur^ nun K r i i

-O la diem tu\ v, ta c6:

if k , G A , + k G A , + + k „ G A , , = 0

k, ( O A , - O G ) + k2 ( O A 2 - O G ) + + k„ ( O A , , ' - O G ) = 0

o O G = ^ ( k , O A , + k 2 0 A 2 + + k „ O A „ ' )

Suy ra G xac d i n h d u y nhat

D A N G T O A N 4: PHAN TICH MOT VECTO THEO HAI

VECTO KHONG CUNG PHUONG

1 PHUONG PHAPGIAI

Sir d u n g cac tinh chat phep toan vecto, ba quy tac phep toan vecto va tinh

chat trung diem, trong tam trong tam giac

2. CAC Vi Dg

Vidu V Cho tam giac ABC Dat a = AB, b = A C

a) Hay d i r n g cac diem M , N thcSa man: A M = - AB, C N = 2BC

3

b) Hay phan tich C M , A N , M N qua cac vec to a va b

c) Goi 1 la diem thoa: M I = C M C h u n g m i n h 1,A,N thcing hang

LOT^ifli (hinh 1.23)

1

a) V i A M = - A B suy ra M thuoc canh A B va

A M = ^ A B ; C N = 2BC, suy ra N thucV tia

Vi du 2: Cho tam giac A B C , tren canh BC lay M sao cho B M = 3 C M , tren

doan A M lay N sao cho 2 A N = 5 M N G la trong tam tam giac A B C a) Phan tich cac vecta A M , B N qua cac vec t o A B va A C

b) Phan tich cac vecto GC, M N qua cac vec to G A va GB

L a i ^ i d i (hinh 1.24)

3 _ 5 • a) Theo gia thiet ta c6: B M = - B C va A N = - A M

= - G A + - G B

2 7

V I du 3: Cho h i n h b i n h hanh A B C D Goi M , N Ian l u o t la hai d i e m n^m tren

hai canh A B va C D sao cho A B = 3 A M , C D = 2 C N v a G la t r p n g tamjtam giac M N B Phan tich cac vecto AN, MN, A G qua cac vec t o ^ A B j ^ a ^ C ^

-;Suy ra A G = — A B + - A C

18 3

3 BAI TAP LUYgN TAP

Bai 1.41: Cho tam giac ABC Lay cac diem M , N , P sao cho M B = 3 M C ,

N A + 2 N C = 6 , PA + PB = d

a) Bieu dien cac vecto A P , A N , A M theo cac vecto A B va A C

b) Bieu dien cac vecto M P , M N theo cac vecto A B va A C

Chung m i n h rang ba diem M , N , P th^ng hang?

Huong dan gidi

a) A P = 1 AB, A N = - A C , A M = - A C - i AB

2 2 2 2

3 - - 1 b) M P = A B - - A C , M N = - A B - - A C

2 2 4

M P = 2 M N => M , N , P th5ng hang

Bai 1.42: Cho tam giac ABC.Goi I , J la hai diem xac d j n h boi

i A = 2IB, 3JA + 2JC = d a) Tinh IJ theo A B va A C

b) Chung m i n h d u o n g th^ng IJ di qua trong tam G ciia tam giac A B C

Hu&ng dan gidi

b) IG = - - A B + i A C = > 5 i j = 6IG suy ra IJ d i qua trong tam G cua tam giac ABC

Bai 1.43 Cho tam giac ABC c6 trong tam G Goi I la diem tren canh BC sao

cho 2CI = 3BI va J la diem tren BC keo dai sao cho 5JB = 2JC

a) Hay phan tich A I , AJ theo AB va A C

34

b) Hay phan tich A G theo A I va A J

Huang dan gidi

3 2 7

a) Ta C O • 2IC = -3IB<=> A I = - A B + - A C «

5 5

5 j ^ = 2JC « 5( A B - AJ) = 2( A C - AJ) « AJ = | A B -1 A C • b) Gpi M la trung diem BC, ta co:

A G = I A M = | | ( A B + AC) = 1( AB + AC) => A~G = ^ A I - : ! AJ

Bai 1.44: Cho hai vecto a, b khong ciing phuong T i m x sao cho ,M a

a) u = a + ( 2 x - l ) b va v = xa + b ciing phuong b) u = 3a + xb va u = ( l - x ) a - - b cung huong

Huong dan gidi

a) u cimg p h u o n g v o i v <» c6 so thuc k sao cho

DANG TOAN 5: CHUNG MINH HAI DIEM TRUNG NHAU,

HAI TAM GIAC CUNG TRONG TAM

1 PHUONG PHAPGIAI

• De chung m i n h hai diem A j va A 2 triing nhau, ta lya chpn mpt trong hai

each sau :

Cdch 1: Chung m i n h A 1 A 2 =0

Cdch 2: C h u n g m i n h O A i = O A j voi O la diem tuy y

De chung m i n h hai tam giac ABC va A ' B ' C cimg trong tam ta lam n h u sau:

Cdch 1: C h u n g m i n h G la trong tam AABC triing voi G ' la trong tam

A A ' B ' C

Cdch 2: Goi G la trong tam AABC (tuc ta c6 G A + GB + GC = 0 ) ta d i chung

m i n h G A ' + GB' + GC' = 0

PJiaii loiii vc'i phutriig phiip ^it'ii IIluli hoc 10

2 CAC VI Dg *

Vt du 1: Chung minh rang AB = CD khi va chi khi trung diem cua hai doan

thang AD va BC trimg nhau

Goi I, J Ian lugt la trung diem cua AD va BC suy ra A I = ID, CJ = JB

Do do AB = CD » A I + IJ + JB = CJ + JI + ID

o i j = Jl o IJ = d hay I triing v6i J , , j, r

Vi du 2: Cho tam giac ABC, tren cac canh AB, BC, CA ta lay Ian lugt cac

NT O , A M BN CP „ , ,

diem M, N, P sao cho = = Chung mmh rang hai tam giac

AB BC CA ^ ^ ^ ABC va MNP c6 cung trgng tam ^ " ^ Lai gidi

A M

Gia su ^ = k suy ra A M = kAB ; BN = kBC ; CP = kCA

Cdch 1: Ggi G, G' Ian lugt la trgng tam AABC va AMNP

Ket hgp vai (*) ta dugc GG' = 0 j

Suy ra dieu phai chung minh

Cdch 2: Ggi G la trgng tam tam giac ABC suy ra G A + GB + GC = 6

Taco: GM + G N + GP = GA + A M + GB + BN + GC + C P

= A M + BN + CP = kAB + kBC + kCA = k( AB + BC + CA) = 0 Vay hai tam giac ABC va MNP c6 cung trgng tam

Vidu 3: Cho luc giac ABCDEF Ggi M , N , P, Q, R, S Ian lugt la trung diem

cua cac canh AB, BC, CD, DE, EF, FA Chung minh rang hai tam giac

MPR va NQS c6 cung trgng tam "'^ ^ '

Lai gidi (hinh 1.26)

Ggi G la trgng tam cua AMPR suy ra GM + GP + GR = 6 (*)

Suy ra G la trgng tam ciia ASNQ '^0^: •

Vay AMPR va ASNQ c6 cimg trgng tam

Vidu 4: Cho hai hinh binh hanh ABCD va AB'C'D' chung dinh A Chung

minh rang hai tam giac B C D va B'CD' cung trgng tam

Lm^ia; (hinh 1.27) Ggi G la trgng tam tam giac BC' D ' , #

Tu (1) va (2) ta CO GB" + GC + GD" = 0 hay G la trgng tam tam giac B' CD'

3 BAI TAP L U Y ^ N TAP

Bai 1.45 Cho cac tam giac ABC, A ' B ' C c6 G, G' Ian lugt la trgng tam Chung

minh rang: A A ' + BB'+ C C = 3GG' Tix do suy ra dieu ki^n can va dii de

hai tam giac c6 cimg trgng tam

Huang dan gidi

Bai 1.46 Cho tam giac ABC, ve cac hinh binh hanh ABIJ, BCPQ, CARS

Chung minh rang ARIP, AJQS c6 ciing trgng tam /

Huang dan gidi

G la trong tam ARIP => GR + GI + GP = 6

Ta C O RJ + IQ + PS = (RA + JA) + (iB + BQ) + (PC + C S )

= RA + CS) + (fA + IB) + (PQ + PC) = 0 Suy ra GR + a + GP = Q + GQ + GS GJ + GQ + GS = d

Do do G la trong tam AJQS

Bai 1.47. Cho tu giac ABCD Goi M, N, P, Q Ian lugt la trung diem ciia AB, BC,

CD, DẠ Chung minh rang hai tam giac ANP va CMQ c6 cung trong tam

Huang dan gidi

G la trong tam AANP => GA + GN + GP = 6 ur,.)

1 -r^ 1 Taco AC + N M + PQ = A C - - A C - i A C = 0

Suy ra G A + G N + GP = GC + G M + G Q ^ G C + GM + GQ = d

Do do G la trpng tam ACMQ

Bai 1.48. Cho tam giac ABC Goi Á, B' ,C' la cac diem xac djnh boi

2011A7B + 2012ÁC = 0, ZOllB^C + 2012B' A = 0 , 2011C A + 2012CB = 0

Chung minh rang AABC va A A ' B ' C cung trong tam

Hucmg dan gidi

G la trong tam AABC =:> GA + GB + GC = 0

Taco 2 0 n A ^ + 2012A7c = 0

o 2 0 1 I ( A ' A + A B ) + 2012(A7^ + AC) = 0

c:>4023ÃA + 20nAB + 2012AC = 0 (1)

Tuong t u ta CO

4023B' B + 201IBC + 2012BA - 0 ; 4023C'(: + 20nCA + 2012CB = 6

Cpng ve vai vélai ta dugc

4023( A A ' + BB' + CC') + BA + AC + CB = O o A A ^ + BB' + C C ' = 0

Suy ra GA + GB + GC = GẤ + GB' + GC' => GAT' + GB' + GC"' = 0

Do do G la trong tam A A ' B ' C

Bai 1.49 Cho AABCva A A ' B ' C c 6 ciing trong tam G, ggi Gi,G2,G3la trong

tam cac tam giac BCÁ,CAB', A B C Chung minh rang AGiG2G3Cung c6

trpng tam G

Huang dan gidi

Vi AABC va A A ' B ' C CO ciing trong tam G suy ra AA" + BB' + C C = 0

Suy ra A G , + BG2 + CG3 =0 do do G la trong tam AG,G2G3

Bai 1.50 Cho t u giac ABCD c6 trong tam G Goi G], G2, G3, G4 Ian lugt la

trgng tam cac tam giac AABC, ABCD, ACDA, ADAB Chung minh r^ng G

cung la trgng tam t u giac G,G2G3G4

Huang dan gidi

tam giac ABC va A,B,C, c6 ciing trgng tam

Huang dan gidi

Ggi D , E, F tuong ung la giao diem ciia M A , , MB,, M C , voi cac canh BC,

CA, AB O la trgng tam deu AABC

Ta C O MA," + MB,' + MC, = 2 ( M D + ME + M F '

r r ; ; 3;

Mat khac theo bai tap 6 (dang 2) thi wb + ME + MF = ^ M O

Suy ra M A , + MB, + MC, = 3MO do do O la trgng tam tam giac A,B,C,

Bai 1.52. Cho tam giac ABC, diem O nam trong tam giac Ggi A , , B i , C , Ian

lugt la hinh chieu ciia O len BC, CA, AB Lay cac diem A2,B2,C2 Ian lugt thugc cac tia O A , , OB,, OC, sao cho OA2 =a, OB2 = b, OC2 = c Chung minh O la trgng tam tam giac A2B2C2

Huang dan gidi

Ta CO OA2 + OB2 +OC2 = ^ O B , OC,

= + b ? ? ^ + = 0 (Theo djnh ly con nhim)

OA, OBi O C , - +

c-Do do O la trgng tam tam giac A2B2C2

39

DANG TOAN 6: T/M TAP HOP DIEM THOA MAN DIEU

KIEN VECTO CHO TRUOC

1 PHLfONG PHAPGlAl

De tim tap hap diem M thoa man man dieu ki?n vecto ta quy ve mot trong

cac dang sau:

- Ne'u M A = MB voi A, B phan biet cho tnroc thi M thuoc duong trung true

aia doan AB

voi A, B, C phan biet cho triroc thi M thuoc duong tron

- Neu MC =k A B

tarn C, ban kinh bang k AB

- Ne'u M A = kBC voi A, B, C phan bi^t va k la so' thuc thay doi thi ^

+ M thuoc duong thang qua A song song voi BC voi k e R

+ M thuoc nira duong thang qua A song song voi BC va ciing huong BC

voi k > 0

+ M thuoc nua duong th^ng qua A song song voi BC va ngu-oc huong BC

vol k < 0

'•i Ne'u M A = kBC, B^C voi A, B, C thang hang va k thay doi thi tap hop

diem M la duong thang BC

2 CAC Vl DU

Vidu 1: Cho tam giac ABC

a) Chung minh rSng ton tai duy nha't diem I thoa man : 2iA + 3IB + 4iC = 0

b) Tim quy tich diem M thoa man : |2MA + 3MB + 4MC| = |MB - M A

Liri gidi

a) Taco: 2IA + SIB + 4iC = 0<=>2iA + 3(IA + i^) + 4(iA + AC) = 6

o 9IA = -SAB - 4AC <^ lA 3AB + 4ACI > I ton tai va duy nha't

b) Voi I la diem duoc xac djnh 6 cau a, ta c6:

2MA + 3MB + 4MC = 9Mi + (2IA + 3IB + 4IC) = 9MI va MB - M A = AB nen

12MA + 3MB + 4MCIHMB - M A l o l 9Mi W AB i o M I = ^

Vay quy tich ciia M !a duong tron tarn I ban kinh A B

Vidu 2: Cho tam giac ABC Tim tap hop cac diem M thoa man dieu kien sau:

a) | M A + MB| = | M A + MCl

b) M A + MB = k ( M A + 2MB - 3MC) voi k la so thuc thay doi

40

Lai gidi (hinh 1.28)

a) Goi E, F Ian luot la trung diem ciia AB, AC suy ra SJA + MB = 2ME va MA + MC = 2MF

Khi do I M A + M B = M A + M C

| 2 M F | < » M E = M F 2ME

Vay tap hop cac diem M la duong trung true ctia EF

Vi du 3: Cho tu giac ABCD Voi so k tuy y, lay cac diem M va N sao cho

AM = kAB, D N = kDC Tim tap hop cac trung diem I cua doan thing

MN khi k thay doi

LOT^ifli (hinh 1.29) Goi O, O' Ian lugt la trung diem ciia AD va BC, ta c6

AB = Ad + OO" + O'B va DC = DO + OO' + O C Suy ra A B + DC = 2 0 0 '

Tuong tu vi O, I Ian luot la trung diem ciia AD

va M N nen A M + D N = 20i

Dodo Oi = i ( k A B + kDC) = kOO' 1 Hinh 1.29

Vay khi k thay doi, tap hop diem I la duong thang OO'

3 BAI TAP LUYlN TAP

Bai 1.53 Cho 2 diem co djnh A , B Tim tap hop cac diem M sao cho:

a) I M A + M B I = I M A - M B I b) |2MA + MB| - | M A + 2MB|

Huang dan gidi

3) Tap hop diem M la duong tron tam I ban kinh voi I la trung diem cua

Ta c6: 2 M A + M B MB + 2MC « MK M L

Tap hop diem M la duang trung true cua doan thang KL

Bai 1.54 Cho AABC Tim tap hop cac diem M sao cho:

a) M A + kMB = kMC voi k la so thuc thay doi

b) V = MA + MB + 2MC cung phirong vol vec to BC

c) I M A + BCl = | M A - MB| ( H D : dung hinh binh hanh ABCD)

Huong dan gidi

Huang dan gidi

a) Goi K la diem thoa man: 2KA + 3KB = 0

L la diem thoa man: 3LB - 2LC = 0

Ta c6:

2 M A - M B - M C

2MA + 3MB = 3MB + 2M(: o MK = ML

=> Tap hop diem M la duong trung true cua doan thSng KL

b) Voi I la trung diem ciia BC Goi J la diem thoa man: 4JA + JB + JC = 6

Ta c6: 4MA + MB + MC 2 M A - M B - M C

6MJ 2 M A - 2 M I 6MJ 2IA; <^MJ = - I A 1

3 Vay tap hop diem M la duong tron tarn J ban kinh R = - lA

3

Bai 1.56: Cho iu giac ABCD

a) Xac dinh diem O sao cho : OB + 40C = 20D ,

b) Tim tap hop diem M thoa man he thuc I M B + 4MC - 2Mi3 = 3MA

Huang dan gidi

a) OB + 40C = 20D <r> OB = - C I vol I la trung diem BD

3

b) M B + 4 M C - 2 M D = 3 M A | I M O = M A Vay tap hop diem M la duong trung true ciia doan O A Bai 1-57: Cho luc giac deu A B C D E F Tim tap hop cac diem M sao cho :

M A + M B + M C + M D + M E + M F nhan gia tri nho nha't

Huang dan gidi

Ggi P la trong tarn ciia A A B C , Q la trong tam cua A D E F

.Ml":

MA + MB + MC MD + ME + MF = 3 M P + 3 M Q = 3 ( M P + M Q ) > 3 Q P

Dau " = " xay ra khi va chi khi M thuoc doan P Q Vay tap hop cac diem M can tim la moi diem thuoc doan P Q Bai 1.58: Tren hai tia Ox va Oy ciia goc xOy lay hai diem M , N sao cho

OM + O N = a voi a la so' thuc cho truoc tim tap hop trung diem I ciia doan thang M N

Huang dan gidi

Goi hai diem M„, N „ Ian lugt thuoc tia Ox va Oy sao cho OMQ = ON(, = - Gia sir O M = k, 0 < k < a khi do ta c6 M I = ^ ^ M o N ( , '

2 a

Do do tap hop diem I la doan MQNQ

DANG TOAN 7: XAC DINH TINH CHAT CUA HINH KHI

BIETMOTDANG THUC VECTO

1 PHLTONG P H A P G I A I

Phan tich dugc djnh tinh xua't phat tir cac dSng thirc vecto ciia gia thiet, luu

y toi nhirng he thuc da bie't ve trung diem ciia doan thang, trong tam ciia tam giac va ket qua " ma + nb = 0 <=> m = n = 0 voi a, b la hai vecto khong

2 CAC VI DU

Vi du 1: Goi M , N Ian lugt la trung diem cua cac canh AD va DC ciia tir giac

ABCD Cac doan th^ng A N va BM ck nhau tai P Bie't ?U = ^BM;

= SAP - 4 A M = 2AN - 2AD

= 2(AD + DN) - 2Al3

= 2DN = DC =i> ABCD la hinh binh hanh

Vi du 2: Cho tam giac ABC c6 cac canh bang a, b, c va trong tarn G thoa

man:a^GA + b^GB + c^GC = O.Chung minh rang ABC la tarn giac deu

Vi du 3: Cho tam giac ABC c6 trung tuyen AA' va B' , C la cac diem thay

doi tren CA, AB thoa man XK' + BB' + CC' = 0 Chung minh BB', CC la

cac trung tuyen cua tam giac ABC

Led gidi

Gia six AB' = mAC, AC' = nAB

Suy ra BB' = AB' - AB = mAC - AB

va CC" = AC' - AC = nAB - AC

Mat khac A' la trung diem ciia BC nen A A ' = ^ ( A B + Ac)

Dodo A A ' + BB' + C C = d <::>-(AB +AC) + m A C - A B + n A B - A C = 0

f 1^ n — AB + m — AC = 0 f 1^

I 2) I 2)

Vi AB, AC khong cung phuong suy ra m = n = - do do B', C Ian lugt la

trung diem ciia CA, AB

Vay BB', CC la cac trung tuyen ciia tam giac ABC

3 BAI T A P L U Y ^ N TAP

Bai 1.59: Cho t u giac ABCD c6 hai duong cheo cat nhau tai O thoa man

OA + OB + OC + OD = 0 Chung minh tu giac ABCD la hinh binh hanh

Cty TNHH MTV DWH Kliang Viet

Huang dan gidi

Dat OA = xAC, OC = yAC, OB = zBD, OD = tBD Suy ra 6A + OB + OC + OD = 0 » ( x + y)AC + (z + t)BD = 0

Do do X = - y ; z = - t => OA = OC, OB = OD nen tu giac ABCD la hinh binh hanh

Bai 1.60: Cho ABC c6 BB', CC la cac trung tuyen A' la diem tren BC thoa man

A A + B B ' + O C = 0 Chung minh AA' cung la trung tuyen ciia tam giac ABC

Huang dan gidi

Huong dan gidi

Gia su A'B = kA'C, B ^ = mB^A, C A = nC'B

A ^ = kA~C <^ AB - A A ' = k(AC - A A ' ) A A ' = Tuang tu ta c6

AD + BC = 2 f i Chung minh J la trung diem ciia CD ^^^^

Huang dan gidi

AD + BC = 2ij<:^ID + IC = 2IJ Goi K la trung diem DC suy ra 10 + 10 = 2 ^ do do K = J hay J la trung diem ciia CD

Bai 1.63: Cho ti> giac A B C D Gia sit ton tai diem O sao cho OA = OB = OC = OD

va O A + OB + OC + O D = d Chii-ng m i n h rang A B C D la hinh c h u nhat

Huang dan gidi Goi M , N, P, Q la trung diem cua AB, BC, CD, D A A

Tit p h u o n g trinh t h u hai ta duoc:

<=> N , Q , 0 thang hang va O la trung diem N Q

Ta CO A O A D can tai O nen N Q 1 A D , AOBC can tai O

nen N Q 1 BC suy ra A D / /BC

T u a n g t u A B / / D C suy ra A B C D la hinh binh hanh

M a N , Q la trung diem ciia BC, A D nen A B / /NQ => A B 1 BC

Suy ra A B C D la hinh chu nhat

Bai 1.64: Cho tam giac ABC noi tiep d u a n g tron tam O, goi G la trong tam tam

giac ABC A', B', C la cac diem thoa man:

O A = 3 0 A ' , OB = B O B O C = 3 0 C '

C h u n g m i n h rang G la true tam tam giac A ' B ' C

Huang dan gidi

G la trong tam tam giac ABC nen 3 0 G = O A + OB + O C

Do do OG = O A ' + OB' + 6C'

Suy ra G la true tam tam giac A ' B ' C

Bai 1.65: Cho tam giac ABC noi tiep d u a n g tron tam O, goi H la true tam tam

giac A', B', C la cac diem thoa man: O A ' = 3 0 A , OB' = 3 0 B , OC"' = SOC

C h u n g m i n h rang H la trong tam tam giac A ' B ' C

Huang dan gidi

H la true tam tam giac ABC suy ra O H = O A + OB + OC

Do do 3 0 H = O A ' + OB' + O C hay H la trong tam tam giac A ' B ' C

Bai 1.66: Cho tam giac ABC va diem M nam trong tam giac D u a n g th^ng A M

cat BC tai D, B M cSt C A tai E va C M c3t AB tai F C h u n g m i n h rling neu

A D + BE + CF = 0 thi M la trong tam tam giac A B C

Huang dan gidi

Truoc tien ta chung m i n h bo de sau

Cho ba vec to a; b;c doi mot khong cung p h u o n g va thoa man dieu kien:

ma + nb + pc = 0 ( ^ , ^ , ^ 0 )

m'a + n ' b + p ' c = 0 ft,

Chung m i n h rSng • = A = • That vay :

p i thay m , m ' * 0 thi suy ra ngay n, n', p, p' cung phai khac khong

Tro lai bai toan

— _ A n • RF • CF • Taco A D + BE + CF = O o i l i i M A + — M B + — ^ M C = 0 Mat khac

1 1 1

S - S S - S , S-Sb

Ap d u n g bo de suy ra ' = —

trong tam tam giac ABC

^ e> Sg = Sb - S^ hay M trung

II

.Pi

DANG TOAN 8: CHUNG MINH BAT DANG THUC VA TIM

cue TBI LIEN QUAN DEN DO DAI VECTO

• P H U O N G PHAP

Su d u n g bat d§ng thuc ca ban:

Voi m o i vecto a, b ta luon c6:

+ +

a + b

a - b

+ , dau bSng xay ra khi a, b cijng h u o n g

, dau bang xay ra khi a, b nguoc h u o n g

47

fnan loai va phuarng phap giai Hinh hoc 10

• Dua bai toan ban dau ve bai toan tim cue tri cua M I voi M thay doi

+ Neu M la diem thay doi tren duang thang A khi do M I dat gia tri nho

nha't khi va chi khi M la hinh chieu ciia M len A

+ Neu M la diem thay doi tren duang tron (O) khi do M I dat gia trj nho

M I nhat khi va chi khi M la giao diem ciia tia OI vai duang tron;

Ian nha't khi va chi khi M la giao diem ciia tia lO vai duang tron

2 CAC VI Dg

dat gia trj

Vt du 1 Cho tam giac ABC va duong thSng d Tim diem M thuoc duong

thang d de bieu thuc sau dat gia trj nho nhat T = M A + M B - M C

Lai giai:

Gpi I la dinh thu tu ciia hinh binh hanh ACBI thi lA + IB - IC = 0

Khi do : T = ( M I + I A ) + ( M I + IB) - (M I + I C )

M i + IA + I B - i C M l Vay T dat gia trj nho nha't khi va chi khi M la hinh chieu ciia I len duang

thang d

Vidu 2: Cho tam giac ABC va A ' B ' C la cac tam giac thay doi, c6 trong tam

G va G' co'djnh Tim gia trj nho nha't ciia tong T = AA'+ BB'+ C C

Dling thuc xay ra khi va chi khi cac vecta Ah', BB", C C ciing huang

Vay gia trj nho nha't T la 3GG'

3 B A I T A P L U Y ^ N T A P

Bai 1.67: Cho tam giac ABC, duang thang d va ba s6'a,p,y sao cho

a + p + y ^ 0

Tim diem M thuoc duang thSng d de bieu thuc T = a M A + PMB + yMC dat

gia trj nho nha't

Huang dan giai

po a + P + y O nen ton tai duy nhat diem I sao cho alA + piB + ylC = 0

Ta C O a M A + pMB + yMC = a(MI + IA) + P(Mi + IB) + y(MI + IC)

= (a + P + y)MI + aIA + pIB + yIC =(a + p + y)MI

Do do T = a + p + Y -MI Suy ra T nho nha't khi va chi khi M la hinh chieu ciia I len duong thing d

Bai 1.68: Cho tam giac ABC Tim diem M tren duong tron (C) ngoai tie'p tam

giac ABC sao cho | M A + MB + MC

b) Dat gia trj nho nhalt '

Huang dan giai

"' i M G

a) Dat gia tri Ion nha't

G la trong tam tam giac ABC ta c6 MA + MB + MC = 3 a) M la giao diem ciia tia GO vai (C)

b) M la giao diem ciia tia OG vai (C)

Bai 1.69: Cho tam giac ABC M, N, P Ian lugt la cac diem tren cac canh BC,

CA, AB sao cho BM = kBC, CN = kCA, AP = kAB Chung minh rang cac doan thing AM, BN, CP la ba canh ciia mot tam giac nao do

Huong dan giai

Taco A M + B N + CP = 0

Suy ra A M = - ( B N + CP): A M B N + CP B N CP

Vi BN va CP khong cung phuong nen khong the xay ra dau bang do do

A M < BN + CP Tuang tu ta c6 BN < A M + CP, CP < A M + BN

Bai 1.70: Cho tam giac ABC Chung minh rang voi mpi diem M thuQC c^nh AB

va khong triing vai cac dinh ta c6: MCAB < MA.BC + MB.AC ^

Huang dan giai

C M : ' ^ - C A M A c B : , M C < M ^ C A M A C B

AB

AB AB ' AB Hay M C A B < MA.BC + MB.AC

Bai 1.71: Cho tu giac ABCD, M la diem thuoc doan CD Goi p, pi, P2 la" ^^^t

la chu vi ciia cac tam giac AMB, ACB, ADB Chung minh rang p < max (pi; P2} •

Huang dan gidi

Bai 1.72: Tren duong tron tam O ban ki'nh bang 1 lay 2n + l diem

Pj, i = l,2, ,2n + l ( n e N) a cung phia vol doi voi duong kinh nao do

1 - - •

Chung minh rang |2n+l

i=l > 1

Huang dan gidi

Ta chung minh bang quy nap

+ Voi n = 0 : hien nhien

+ Gia su BDT diing vol n = k ta di chung minh dung voi n = k +1 hay

§4 TRUC TOA DO VA HE TRUC TOA DO

^ TOM TAT LY THUYET , j 4

I T R V C T O A D Q :

I Dinh nghia: True toa do (True , hay true so ) la mot duong thSng tren do ta

da xac dinh mot diem O va mot vecto don vi i (tue la i = 1)

Hinh 1.30

Diem O dugc goi la goc toa do, vec to f dugc gpi la vecta dan vi cua true toa

dp. Ki hieu (O ; i ) hay x'Ox hoae don gian la Ox ;

2 Tpa dp cua vecto va cua diem tren true:

+ Cho vec to u nam tren true (O ; f ) thi c6 so thuc a sao cho u = a l voi

a e R So a nhu the dupe goi la toa dp eiia vecto u" doi voi true (O ; i')

+ Cho diem M nam tren ( O ; i ) thi eo so m sao cho O M = m i So'm nhu the dupe goi la toa dp cua diem M doi voi true ( O ; i )

Nhu vay tpa dp diem M la tpa dp vecto O M

3 Dp dai dai so ciia vec to tren true:

Cho hai diem A, B nam tren true Ox thi tpa dp cua vecto AB ki hi|u la AB

va gpi la dp dai dai so cua vecto AB tren trye Ox , Nhu vay AB = AB.i'

Tinhchat: ' * + A B = - B A

+ AB = C D » A B = CD + V A ; B ; C e ( 0 ; T ) : AB + BC = AC

Mat khae theo gia thiet quy nap ta eo O B '2k+2 ZOP,

I , ] Diem O gpi la goc toa do, Ox gpi la true

hoanh va Oy gpi la true tung

P A + O B O B >1 Kihieu Oxy hay ( 0 ; i , j ) Hinh 1.3

2- Tpa dp diem, tpa dp vec to , Trong he true tpa dp (0;T,]) neu u = xT + yj thi cap so (x;y) dupe gpi la tpa

dp eiia vecto u , kihieu la u = (x;y) hay u ( x ; y ) I f-\ ' •

S I

X dirgrc goi la hoanh do, y duQC gpi la tung dp ciia vecto u

+ Trong true tpa dp (0;i,j), tpa dp cua vecto OM gpi la tpa dp cua diem

M, ki hi?u la M = (x;y) hay M(x;y) x dupe gpi la hoanh dp, y dupe gpi la

tung dp cua diem M * #i>i,'^%t ' "

Nhqn xet: (hinh 1.31) Gpi H, K Ian lupt la hinh chieu eiia M len Ox va Oy

thi M(x;y)<:>OM = xr+y] = 6 H + OK % C

Nhu vay O H = xi, OK = y] hay x = OH, y = OK

3 Tpa dp trung diem cua doan thang Toa dp trpng tarn tam giac

+ Cho M^A'VA)' ^ X B ' Y B ) va M la trung diem AB Tpa dp trung diem

YM =

2 ' 2

+ Cho tam giac ABC c6 A(xy^;y;i^), B(xB;yB), C ( x ( ; ; y c ) Tpa dp trpng tam

G ( x c ; y G ) c u a t a m g i a e A B C I a x c = ^ ^ ^ ^ ^ ^ va y c = ^ ^ ^ ^ ^ ^

4 Bieu thurc tpa dp cua cac phep toan vecto

Cho u =(x;y) ; u ' = (x';y') va so thuc k Khi do ta c6 :

fS'-_ fS'-_ fx = x'

1) u = u'<=>^

l y = y 2) u ± v = ( x ± x ' ; y ± y ' )

3) k.ii = (kx;ky) ^

4) u* Cling phuong u (u 7t 6) khi va chi khi c6 so k sao cho I ,

y - k y

5) Cho A(xA;yA), ^^BiYs) A B = ( X B - X A ; y B - Y A )

DANG TOAN 1: T/M TOA DO CUA MOT DIEM; TOA DO

VECTO; DO DAI DAI SO CUA VECTO VA CHUNG MINH

HE THUC UEN QUAN TREN TRUC (0; 1)

1 P H U O N G P H A P GlAl

Sir dung cac kien thuc ca ban sau:

• Diem M c6 tpa dp a o OM = a.f '*

• Vecto AB codpdaidaisola m = AB<=> AB = mT

^ Neu a, b Ian lupt la tpa dp ciia A, B thi AB = b - ^

, Cac tinh chat + A B = - B A A, m

a

Vidu 2: Tren true tpa dp (O; i ) cho 4 diem A, B, C, D bat ky

Chung minh AB.CD + AC.DB + AD.BC = 0

Cach 2: AB.CD + AC.DB + AD.BC =

A B ( A D - A C ) + A C ( 7 ^ - A D ) + A D ( A C - A B )

= AB.AD - A B A C + A C A B - A C A D + A D A C - AD.AB

= 0 ^

3 BAI T A P L U Y ^ N T A P ( V

Bai 1.73.Tren true tpa dp (O; f ) Cho 2 diem A va B c6 tpa dp Ian lupt a va b

a) Tim tpa dp diem M sao cho MA = kMB (k ^ 1) ' b) Tim tpa dp trung diem I ciia AB ' ^ \ c) Tim tpa dp diem N sao cho 2NA = -5NB ' ' • ^

k b - a a + b ^ 5b + 2a ?

-Bai 1.74 Tren true toa do (O ; i ' ) cho 4 diem A , B, C, D c6 toa do Ian Ixxqt la

a, h, c, d va thoa man h$ thu'c2(ab + cd) = (a + b)(c + d )

DANG TOAN 2: TIM TOA DO DIEM, TOA DO VECTO

TREN MAT PHANG Oxy

1 P H L T O N G P H A P

• De t i m tpa do cua vecto a ta lam n h u sau

D i m g vecto O M = a Gpi H , K Ian lugt la hinh chie'u vuong goc cua M len

Ox, O y K h i do a ( a i ; a2) voi a 1 = O H , a2 = OK ,

• De tim toa do diem A ta di tim toa do vecto O A

• Neu biet toa do hai diem A ( x A ; y A ) , B(xB;yB) suy ra toa dp A B dupe xac

d j n h theo cong thuc AB = (xg - x ^ ; yB - YA )

Chii y: O H = O H neu H nam tren tia Ox (hoac O y ) va O H = - O H neu H

nam tren tia doi tia Ox (hoac O y )

2 CAC Vl DU:

Vidu 1: Trong mat phSng tpa dp O x y

T i m tpa dp cua cac diem M2*

a) M l doi xung voi M qua true hoanh /

b) M 2 doi xung voi M qua true tung / 0 X

c) M3 doi xung vai M qua goc tpa dp / Hinh 1.32

; V LOT ^/at (hinh 1.32)

a) M l doi xung voi M qua true hoanh suy ra M i (x;-y)

]Vl2 doi xung vol M qua true tung suy ra ( - x ; y ) M3 doi xung voi M qua goc tpa dp suy ra M3 ( - x ; - y )

yi^u 2: Trong he true tpa dp (O; i ; j ), cho hinh vuong A B C D tarn I va eo

A(l;3) Biet diem B thupc true (O; i ) va BC cung huong voi f T i m tpa

dp cac vecto A B , BC va A C ''

Lai gidi (hinh 1.33)

T u gia thie't ta xac djnh dupe hinh vuong tren mat phang tpa dp (hinh ben)

Vi diem A ( l ; 3) suy ra A B = 3, OB = 1

D o d o B(1;0),C(4;0),D(4;3) Vay AB(0; -3), BC(3; 0) va AC(3; -3)

0\

Hinh 1.33

Vi du 3: Trong mat phSng tpa dp Oxy Cho hinh thoi A B C D canh a va

6 ^ = 60" Biet A triang v o i goc tpa dp O, C thupc true Ox va

XB ^ 0,yB ^ 0 T i m tpa dp cac dinh eiia hinh thoi ABCD

Lai gidi (hinh 1.34)

Tir gia thie't ta xac dinh dupe hinh thoi tren mat phSng tpa dp Oxy Gpi I la tarn hinh thoi ta c6

BI = A B s i n B A I = asin30" =

-A I = N/-AB^^^B? = Suy ra

A ( 0 ; 0 ) , B [ ^ ; | ] , C ( a V 3 ; 0 ) , D faVs D

Hinh 1.34

3.BAITAPLUYeNTAP ^ ^ i f - v L

Bai 1.75: Cho hinh binh hanh A B C D c6 A D = 4 va chieu cao ihig v o i canh

A D = 3, B A D = 60° Chpn h? tryc tpa d p (A;T,]) sao cho I va A D ciing

huong, yB > 0 T i m tpa dp cac vecto AB, B C , CD va A C

Huang dan gidi

Ke B H 1 A D => B H = 3; AB = 2V3; A H = Vs

5 5

A(0;0) ;B(73;3) C(4 + V3;3) D(4;0)

AB = {73;3) BC=(4;0)CD = (-73;-3)

AC = (4 + V3;3)

Bai 1.76: Cho luc giac deu ABCDEF Chgn he true toa do (O; i ; j ), trong do

O la tarn luc giac deu , i ciing huong voi O D , j cung huong EC Tinh tga

dp cac dinh luc giac deu , biet canh ciia luc giac la 6

Huang dan gidi

A(-6;0),D(6;0), B(-3;3V3),

:(3;3V3), F(-3;-373), E(3;-3V3

DANG TOAN 3: XAC DINH TOA D O DIEM, VECTO LIEN

QUAN DEN BIEU THUC DANG u + v,u-v, ku

1 P H U O N G P H A P

Dung cong thuc tinh toa dp ciia vectau + v, u - v, k u

Voi u = ( x ; y ) ; u ' = (x';y') va so thuc k, khi do u ± v = (x ± x';y ± y') va

a) Taco A B ( 4 ; 3 ) , AC(6;1) suy ra u = ( 2 ; 5 )

b) Gpi M ( x ; y ) , taco M A ( - 4 - x ; - y ) , M B ( - X ; 3 - y), M C ( 2 - x;l - y) Suyra M A + 2 M B + 3 M C = (-6x + 2;-6y + 9 )

-4

^3'2,

3 BAI T A P LUYfiN T A P Bai 1.77.Cho cac vecto a = (2;0), bT =

a) u = 2a' - 4b + 5c | b) a - 2b + 2u = c

' Huang dan gidi

a) u =:(28;-28) b ) u = ( 0 ; | ) ;

Bai 1.78 Cho ba diem A ( - 4 ; 0 ) , B(-5;0) va C(3;-3)^

a) Tim tpa dp vecto u = AB - 2BC + 3CA Tim diem M sao cho M A + MB + MC = 0 '^^ '

Huang dan gidi

, c = ( 4 ; 6 ) Tim tpa dp vecto u biet

57

DANG TOAN 4: XAC DINH TOA DO CAC DIEM CUA MOT HINH

1 PHUONG PHAP

Dya vao tinh chat cua hinh va sit dung cong thuc

+ M la trung diem doan thang A B suy ra x YM = - Z A + Y B

+ G trong tarn tarn giac ABC suy ra x^ = ^ A + X B + X C ^ ^ y A + y B + Yc

(0:i ) f\ fid t:>f{3 ;E HII

Vidu 1: Cho tarn giac ABC c6 A(2;l), B { - l ; - 2 ) , C ( - 3 ; 2 )

a) Tim toa do trung diem M sao cho C la trung diem cua doan MB

b) Xac dinh trong tam tam giac ABC

c) Tim diem D sao cho ABCD la hinh binh hanh

Vidu 2; Trong mat phang toa do Oxy cho A ( 3 ; - 1 ) , B ( - 1 ; 2 ) va l ( l ; - l ) Xac

dinh toa do cac diem C, D sao cho tu giac A B C D la hinh binh hanh biet I

la trong tam tam giac A B C Tim toa tam O cua hinh binh hanh A B C D

3 BAI TAP L U Y S N TAP

Bai 1.79: Cho ba diem A(3;4), B(2;l), C ( - l ; - 2 ) a) Tim toa do trung diem canh BC va toa do trong tam cua tam giac ABC b) Tim toa do diem D sao cho ABCD la hinh binh hanh

Huang dan gidi

[y = 1 ^ ' Bai 1.80: Trong mat phang toa do Oxy cho A ( 3 ; 4 ) , B ( - 1 ; 2 ) , I ( 4 ; 1 ) Xac djnh toa do cac diem C, D sao cho tu giac A B C D la hinh binh hanh va I la trung diem canh C D Tim toa tam O ciia hinh binh hanh A B C D

Huang dan gidi

Do I ( 4 ; - 1 ) la trung diem ciia C D nen dat I

C ( 4 - x ; - l - y ) , D ( 4 + x ; - l + y ) ^ C D ( 2 x ; 2 y )

Vay C ( 2 ; - 2 ) , D ( 6 ; 0 ) , O "

Tu giac ABCD la hinh binh hanh <=> CD = B A <=> •

59

Bai 1 8 1 : Cho tam giac A B C c6 A ( 3 ; 1 ) , B ( 1 ; - 3 ) , d i n h C nam tren Oy va tron^

tam G nam tren true O x T i m toa dp d i n h C

\ dan gidi •/

T u g i a thiet ta CO C ( 0 ; y ) , G ( x ; 0 )

G la trong tam tam giac nen x^ + X B + X C = 3 x c

I,;: nnfcTi 'ria

Bai 1.82: Cho tam giac ABC c6 M , N, P Ian luot la trung diem cua BC, C A , AR

, Biet M ( l ; l ) , N ( - 2 ; 3 ) , P ( 2 ; - 1 ) T i m toa do cac d i n h cua tam giac A B C

Huang dan gidi

T a c o M N ( - 3 ; - 4 ) , P A ( X ^ -2;y^ + l ) , M N = PA ^ A ( - 1 ; - 5 )

N la t r u n g diem A C suy ra C ( - 3 ; - l )

M la t r u n g d i e m BC suy ra B ( 5 ; 3 )

Bai 1.83: Cho tam giac ABC c6 A ( 3 ; 4 ) , B ( - 1 ; 2 ) , C ( 4 ; 1 ) A ' la d i e m doi xung

ciia A qua B, B' la diem doi xung ciia B qua C, C la diem d o i x u n g ciia C

qua A

a) T i m toa do cac diem A " , B", C

b) C h u n g m i n h cac tam giac ABC va A ' B ' C c6 cung trong tam

Huang dan gidi

a) A ' la diem d o i x u n g ciia A qua B suy ra B la trung diem cua A A ' do do

va chi k h i c6 so' k sao cho

Chii y: Neu xy 0 ta c6 u ' cung phuong u o — = y

pg- phan tich c ( c , ; C 2 ) qua hai vecto a ( a , ; a 2 ) , b ( b , ; b 2 ) khong cung

phu-ong, ta gia su c = xa + y b K h i do ta quy ve giai he p h u o n g trinh

aiX + b i y = c,

a2X + b 2 y = C2

2 cAC V I D\}

Vtdu 1: Cho a = (1;2), b = ( - 3 ; 0 ) ; c = (-1;3) a) Chung m i n h hai vecto a ; b khong ciing phuomg b) Phan tich vecto c qua a ; b

Lai gidi

a) Taco ^ ^ ^ = > a va b khong cung p h u o n g b) Gia su c = xa + y b Ta c6 xa + y b = (x - 3y;2x)

2 Suy ra • X - 3 y =

2x = 3

X = —

3 ^ c

5 ^ = 9

V I d H 3 ; T r o n g mat phang toa do O x y , cho ba d i e m A { 6 ; 3 ) , B { - 3 ; 6 ) , C(l; 2)

a) Chirng m i n h A, B, C la ba d i n h mot tam giac ^ b) Xac d j n h diem D tren true hoanh sao cho ba diem A , B, D thang hiing

c) Xac d j n h diem E tren canh BC sao cho BE = 2EC jdpCac d j n h giao diem hai ducmg thang DE va A C

6 1

Lai gidi

a) Ta CO A B ( - 9 ; 3 ) , A C ( - 5 ; - 5 ) V i — s u y ra ABva A C khong cung

phuang ^

Hay A, B, C la ba dinh mot tarn giac

b) D tren true hoanh => D(x;0)

Ba diem A, B, D thang hang suy ra AB va A D cimg phuong |y

Mat khac AD(x - 6; -3) do do — = — x = 15 "

3 BAI TAP LUYgN TAP

pai l - ^ - Trong mat phang toa do Oxy cho 4 diem A ( 1 ; - 2 ) , B(0;3), C(-3;4)

v a D ( - l ; 8 ) a) Bo ba trong 4 diem tren bo nao thang hang ^ ^ b) Chung minh AB va AC khong ciing phuang i '

Huang dan gidi

D (O; 3) Tim giao diem ciia 2 duong thang AC va BD

Huang dan gidi

Goi l(x;y) la giao diem AC va BD suy ra A I ;AC cimg phuong va B I ; BD

cimg phuong Mat khac: A I = ( x ; y - l ) , AC = (2;6) suy ra - = 6 x - 2 y = - 2 (1)

2 6

BI = ( x l ; y 3), BD = (1;0) suy ra y 3 the'vao (1) ta c6 x =

-Vay I la diem can tim

(MS Bai 1.86 Cho a = (3;2), b = (-3;1)

a) Chirng minh a va b khong cimg phuong

b ) Dat u = (2 - x)a + (3 + y)b Tim x, y sao cho u ciing phuang voi xa + b va

V a y c o h a i d i e m t h o a m a n M , ( l ; 0 ) , M 2 ( 3 ; 2 ) ^

Bai 1.88 Cho ba d i e m A ( - l ; - l ) , B(0;1), C(3;0)

a) C h u n g m i n h ba diem A , B, C tao thanh mot tam giac

b) Xac d j n h toa do diem D biet D thuoc doan thMng BC va 2BD = 5DC

c) Xac d j n h toa do giao diem ciia A D va BG trong do G la trong tam tam

cimg p h u o n g suy ra ton tai k : BI = k B G y = 1

a i 1 8 9 T i m tren true hoanh diem P sao cho tong khoang each t u P toi hai

diem A va B la nho nha't, biet:

b) A ( 1 ; 2 ) va B ( 3 ; 4 ) a) A ( 1 ; 1 ) va B ( 2 ; - 4 )

Huong dan gidi

a) D i thay d i e m A, B nkm 6 hai phia v o l true hoanh

Ta C O PA + PB > A B D a u bang xay ra <=> A P ciing p h u o n g v o i A B Suy ra ^ = l z l = X p = ^ = > P

3 - 1 " 4 + 2 X p = - = ^ P

Bai 1.90: Cho h i n h b i n h hanh ABCD c6 A ( - 2 ; 3 ) va tam l ( l ; l ) Biet diem

K ( - 1 ; 2 ) n a m tren d u o n g thang AB va diem D c6 hoanh do gap doi hjng

dp T i m cac d i n h con lai ciia hinh binh hanh

Huong dan gidi

I la t r u n g d i e m A C nen C ( 4 ; - l ) Gpi D ( 2 a ; a ) ^ B ( 2 - 2 a ; 2 - a )

Chuyen del: l ^ N G D V N G V E C T O DE G I A I T O A N H I N H H Q C

Phmmg phap chung

De giai mot bai toan tong hop bang phuong phap vecto ta thuong thyc

hien theo cac buoc sau \

Bitac 1: Chuyen gia thiet va ket luan ciia bai toan sang ngon ngi> cua

vecto, chuyen bai toan tong hgrp ve bai toan vecto

Bie&c 2: Su dung cac kien thuc vecto de giai quyet bai toan do

Bu&c 3: Chuyen ket qua bai toan vecto sang ket qua bai toan tong hop

Sau day la mot so dang toan thuang gap

I C H l f N G M I N H BA DIEM T H A N G HANG, Dl/ONG T H A N G D I QUA

D I E M CO D I N H V A DIEM THUQC DlTONG T H A N G CO D I N H

1 PHUONG PHAP GlAl ^"

• E)e chung minh ba diem A,B,C thang hang ta chung minh hai vec to AB va

AC cung phuong, tuc la ton tai so'thuc k sao cho: AB = kAC

• E)e chung minh ducmg thang AB di qua diem c6' djnh ta di chung minh ba

diem A, B, H thang hang voi H la mot diem co'djnh

2 CAC Vi

DU-Vi du 1: Cho hai diem phan biet A, B Chumg minh rang M thuoc duomg

thang AB khi va chi khi c6 hai so' thuc a , p c6 tong bang 1 sao cho:

• Neu O M = aOA + pOB voi a + p = l=>p = l - a

=> ^ = aOA + (1 - a)OB => O M - OB = a(OA - OB) => BM = aBA

Suy ra M , A, B thSng hang

Vidu 2: Cho goc xOy Cac diem A, B thay doi Ian luot nam tren Ox, Oy sao

cho OA + 20B = 3 Chung minh rang trung diem I cua AB thuoc mot

duomg thang co dinh

D i n h httomg: Ta c6 h? thuc vecto xac djnh diem I l a OI = |oA + ^OB (*)

Tir vi du 1 ta can xac dinh hai diem co djnh A', B' sao cho OI = aOA" + pOB"

3 3 diem A'va B'sao cho O A ' = O B ' = -

i rnfjrt) f i i

Lot gtat

3 3 Tren Ox, Oy Ian lugrt lay hai diem A", B' sao cho OA' = - , OB' = -

Do do diem I thuQC duong thang A'B' co djnh

Vidu 3: Cho hinh binh hanh ABCD, I la trung diem ciia canh BC va E la

AE 2

diem thuoc doan AC thoa man — = T • Chung minh ba diem D, E, I

thing hang

Djnh huong: De chung minh D, E, I thMng hang ta di tim so k sao cho

DE = k D I , muon vay ta se phan tich cac vecto DE, DI qua hai vecto khong

Cling phuong AB va AD va su dung nhan xet " ma + nb = 0 <» m = n = 0 voi a, b la hai vecto khong ciing phuong" tu do tim dugc k = -

Lai giai (hinh 1.35) Taco D I = DC + a = DC + icB = A B - i A D (1)

Vi du 4: Hai diem M, N chuyen dpng tren hai do^n th^ng co'dinh BC va BD

Do cac diem B, H co'dinh, nen diem I co djnh.(xac djnh bai he thuc (3))

Vi du 5: Cho ba day cung song song AAj,BBi,CCi cua duong tron (O)

Chung minh rMng true tam ciia ba tam giac ABCpBCApCABj nam tren

mgt duong thgng

GQI H J , H 2 , H 3 Ian lugt la true tam aia cac tam giac ABC,, BCAj,CABj

Ta co: OH, = OA + OB + OCj, OHj = OB + OC + OA,

va OH3 =OC + OA + OBj

Suy ra H^Hj = OH2 - O H , = O C - O C , + OA7 - O A = C,C + A A ^

\3 = OH3 - OH, = OC - OC,' + OB, - OB = CjC + BBj

Vi cac day cung A A,, BBj, CC, song song voi nhau

Nen ba vecto AA,,BBi,CC,^ co cimg phuong

Do do hai vecto HjHj va HjHg cung phuong hay ba diem Hi ,H2,H3

thang hang

3 B A I T A P L U Y ^ N T A P

, j i^gi: Cho tam giac ABC Gpi M la diem thupc canh AB, N la diem thupc

1 3 c?nh AC sao cho AM = - AB, AN = - AC Gpi O la giao diem cua CM va

— >>•> !m»h BUD iij

BN Tren duong thang BC lay E Dat BE = xBC { li ) fjni X de A, O, E thing hang

Huang dan gidi

Xaco: AO = ^ A B + ^ A C ; AE = (1 - x)A^ + xAC

A, E, O thMng hang <=> AE = kAO , ^ p

o ( l - x ) A B + xAC = - A B + - A C » k = — ; x = —

Vay x = — la gia trj can tim ^ orlD :dQ

1.3 Bai 1.92: Cho AABC lay cac diem I , J thoa man IA = 2IB, 3JA + 2JC = 0

Chung minh rang IJ di qua trpng tam G ciia AABC v • •'—^

-Huang dan gidi ,.;

iA = 2 i B « i A - 2 i B = 6

3JA + 2JC = 0 o 3rA + 2ic = Sij

Suy ra 2(1 A + IB + IC) = SIJ <» 610 = SIJ I, J, G thMng hang "" '

Bai 1.93: Cho tam giac ABC Hai diem M, N di dpng thoa man

M N = MA + MB + MC a) Chung minh rang MN di qua diem codjnh

b) P la trung diem cua AM Chung minh rang MP di qua diem co'dinh

Huang dan gidi

3) Gpi G la trpng tam tam giac ABC suy ra

MN = MA + MB + M C ^ M N = GA + GB + ( X + 3 M ^ Suy ra M , N , G th^ng hang hay MN di qua diem co djnh G

b) P la trung diem A M =^ MP = ^ ( M A + M N ) = i ( 2 M A + MB + MC' Gpi I la trung diem BC, J la trung diem AI suy ra 2JA + JB + JC = 0

Do do MP = 2MJ suy ra MP di qua diem co djnh J

^ai 1.94: Cho hai diem M, P la hai diem di dpng thoa man |

MP = aMA + bMB + cMC

Chung minh rSng MP di qua diem co djnh

69

Huong dan gidi

GQI I la tam duong tron npi tiep tarn giac ABC suy ra alA + bIB + cIC = 0

Do do MP = aMA + bMB + cMC <=> MP = (a + b + c)Mi ,,, ,,,

Vay MP di qua diem CO djnh I i •

Bai 1.95 Cho hinh binh hanh ABCD Goi E la diem dol xung cua D qua die'm

A, F la diem do'i xung cua tam O cua hinh binh hanh qua diem C va K la

trung diem ciia doan OB Chung minh ba diem E, K, F thang hang va K la

trung diem cua EF

Huang dan gidi * *

— s — ' 3 — — s — — HnfciilO 3

Taco: EF = - A D + - A B , EK = - A D + - A B

_ 2 4 4

r^EF = 2EK Vi vay K la trung diem EF. V' ,

Bai 1.96: Cho hai tam giac ABC va A j B j C j ; A 2 , B 2 , C 2 Ian luot la trong tam

cac tam giac BCAj, CABj, ABCj Goi C G p G j Ian lugt la trong tam cac

tam giac ABC, A j B j C j , A j B j C j ^

G G i Chung minh rang C G p G j thang hang va tinh

G G j ;

Huang dan gidi

• Vi G, Gj la trong tam tam giac ABC, AjB^Cj suy ra

Mat khac A A 2 ' + BB^ + C q = AA^ + BB^ + C q

Ma A 2 , B 2 , C 2 Ian lugt la trong tam cac tam giac BCAj, CABj, ABCj

pai 1 97- ^^"^ 8'^*^ • diem M, N, P Ian lirot nam tren duong thSng

B C C A , A B sac cho MB = aMC, NC = PNA, PA = yPB

Tim dieu kien cua a, p, y de M, N, P th^ng hang

Huang dan gidi

Huang dan gidi

Gpi P, Q, R, S Ian lugt la cac tiep diem ciia cac doan thang AB,BC,CD,DA doi v6i duong tron tam O

(b + d)(OA + OC) + (a + c)(OB + O D ) = 0 J

o (b + d)OM + (a + c)ON = 6

Suy ra O, M, N thMng hang (dpcm) Bai 1,99: Cho luc giac ABCDEF noi tiep duomg tron tam O thoa man

AB = CD = E F Ve phia ngoai luc giac dung cac tam giac AMB, BNC, CPD, DQE, ERF, FSA dong dang va can tai M, N, P, Q, R, S Goi O , , O j Ian lugt

la trong tam tam giac MPR va NQS Chung minh rang ba dieim O, O j , O 2 thang hang

71

Huang dan gidi

GQ'I M I , N J , P J , Q , , R I , S , Ian lug-t la hinh chieu ciia M , N , P , Q , R , S len

A B , B C , C D , D E , E F , F A Suy ra M j , N j , P j , Q j , R j , S j Ian lugt la trung diem

cua A B , B C , C D , D E , E F , F A

T a c o MS + R Q + P N = ( M M ^ + M^A + AS^ + S ^ ) + •

+ (RR^ + R j E + EQ 7 + Q i Q : ' = 2 (M M ^ + PPI

• De chung minh duong thSng A B song song voi C D ta di chung minh

A B = k C D va diem A khong thuoc duong thang C D

• De chung minh ba duong thang dong quy ta c6 the chung minh theo hai

huang sau:

+ Chung minh moi duong thang cung di qua mot diem co'djnh

+ Chung minh mot duong thang di qua giao diem ciia hai duong thang

con lai

2. CAC vl og

Vi du 1: Cho ngu giac A B C D E Goi M, N, P, Q Ian lugt la trung diem ciia

cac canh AB, B C , C D , D E Goi I, J Ian luot la trung diem ciia cac doan MP

va NQ Chung minh rkng IJ song song voi A E

Suy ra IJ song song voi A E

yf4u2: Cho tam giac ABC.Cac diem M, N, P thupc cac duong thSng BC,

CA, AB thoaman a + p + y ^ O , + yMC = y N C + a N A = a P A + pP'B = 0 thi A M , B N , C P dong quy tai O, voi O la diem dugc xac dinh boi

a O A + pOB + yOC = 6 ^^^^

Lcngidi ^ iJJ SAl

Ta CO pMB + yMC = 6 c> p (MO + OB ) + y (MO + O C ) = 0 , |., „,,, a O A + pOB + y O C + (p + y) M O = a O A ' ^-^

^ <=>(p + y ) M d = a O A ' • Suy ra M , O, A thang hang hay A M di qua diem co djnh O Tuong tu ta c6 B N , C P di qua O , '

Vay ba duong thang A M , BN, C P dong quy

Vi du 3: Cho sau diem trong do khong c6 ba diem nao thang hang Goi A la

mot tam giac c6 ba dinh lay trong sau diem do va A' la tam giac c6 ba dinh con lai Chung minh rang voi cac each chon A khac nhau cac duong thang noi trong tam hai tam giac A va A' dong quy

Dinh huang Gia su sau diem do la A, B, C, D, E, F ,,,,, V> ; '

Ta can chung minh ton tai mot diem H co dinh sao cho voi cac each chon A khac nhau thi H thuoc cac duong thang noi trong tam hai tam giac A va A' Neu A la tam giac ABC thi A' la tam giac DEF Goi G va G Ian luot la trong tam cua tam giac ABC va tam giac DEF

H thuoc duong thang G G ' khi co so thuc k sao cho H G = k H G

Vi vai tro cua cac diem A, B, C, D, E, F trong bai toan binh dSng nen chpn k

sao cho — = - o k = -1 khi do H A + H B + H C + H D + H E + H F = 0

73

(*) « 3HG + GA + GB + GC = 3HG' + G D + G'E + G T

« H G = H G '

i Do do GG' di qua diem c6' djnh H do do cac duong thang nol trong tam hnj

tam giac A va A ' dong quy

3 B A I T A P L U Y ^ N T A P

Bai 1.100: Cho t u giac ABCD, goi K, L Ian luot la trong tam cua cac tam gia^

ABC va tam giac BCD Chung minh rang hai duong thing KL va A D song

song vai nhau

tuong ung song song voi cac canh ciia tam giac ABC

Huang dan gidi

^

k ^ - k + l

( k l ) ^ A C , v i k ^ - k + l > 0 va A j i ^ A C nen A j C j Z / A C

Tuong tu ta c6 B2C2 / /BC va A j B j / /AB

Bai 1.102: Tren duong tron cho nam diem trong do khong c6 ba diem nao

thang hang Qua trong tam ciia ba trong nam diem do ke duong thing

vuong goc voi du-ong thang di qua hai diem con lai Chung minh rang muoi

duong thang nhan duoc cat nhau tai mot diem

Huang dan gidi

Gia su nam diem do la A j , A j , A 3 , A 4 , A5 nlim tren duong tron (O) Ta

can chung minh ton tai diem H thuoc muoi duong thing do

Goi G la trong tam ciia tam giac A J A 2 A 3 ; P la trung diem ciia doan thing

A4A5 Vi OP 1A4A5 (do OA4 =OA5) nen diem H thupc duong thing di

qua G va vuong goc vol duong thSng A4A5 khi c6 so' thuc k sao cho

J5G = kOP Ma OG = -(OA^ + 6A2 + O A 3 j (vi G la trong tam cua tam giac

chon k sao cho — = - <» k = — \ / ,x

Khi do OH = - ( O A ^ + OA^ + OA^ + OA4+OA^)

ilP' ^ ^v-./' • b "n^i fiii','

Hay OH = - OG (G la trong tam cua h? diem [ A J , A 2 , A 3 , A 4 , A 5 } )

3 Bai 1.103 Cho tu giac ABCD noi tiep duong tron (O) Goi M , N , P, Q Ian luot

la trung diem ciia cac canh AB, BC, CD, DA Ke MM', N N ' , PF, QQ' Ian luot vuong goc voi CD, DA, AB, BC Chung to rang bon duong thang MM', NN', PP', QQ' dong quy tai mot diem Nhan xet ve diem dong quy va hai diem I ,

O (I la giao diem ciia MP va NQ)

Huong dan gtat ^

Ta can chiing minh ton tai diem H thuoc duong thang M M ' , NN', PP', QQ'

Vi OP 1 CD (do OC = OD) nen diem H thupc duong thing M M ' khi c6 so thuc it sao cho H M = kOP O :*';

Ma M va P Ian luot la trung diem ciia AB va CD nen :).u> i

Hay 2 0 H = 401 (De thay / la trong tam ciia tu giac ABCD) <=> OH = 20I

Vay H la diem doi xung ciia O qua /

75

Bai 1.104: Cho n a m diem trong do khong c6 ba diem nao thang hang Goi A la

mot tam giac c6 ba d i n h lay trong nam diem do, hai diem con lai xac dinh

mot doan thang 0 C h i i n g m i n h rang v o i cac each chon A khac nhau cae

d u a n g t h i n g no! trong tam tam giac A va trung diem doan th^ng 9 luon di

qua mot diem co djnh

Huang dan gidi

f Goi A , B, C la ba d i n h ciia tam giac A va DE la doan thSng 0 G o i G la

trong tam tam giac A va M la trung diem ciia DE thi v o i diem O tuy y ta c6

O A + OB + OC + O D + OE = 3 0 G + 2 I M , t

Do do G M luon d i qua diem co d j n h O la trong tam he diem A , B, C, D , E

Bai 1.105: Cho tam giac ABC Ba d u o n g thSng x, y, z Ian lugt d i qua A , B, C va

chiing chia doi chu v i tam giac ABC

C h u n g m i n h rang x, y, z dong quy ' ' '>f>i^ 'A fign

Huang dan gidi

Bai 1.106: Cho tam giac ABC, cac d u a n g tron bang tie'p goc A , B, C tuang u n g

tie'p xuc v o i cac canh BC, CA, A B tai M , N , P.Chung m i n h A M , B N , CP ciing

d i qua m o t diem, xac d j n h diem do

Huang dan gidi

Gia su d u o n g tron bang tie'p goc A tie'p xiic BC tai M

Goi B',C' la tie'p diem ciia canh AB, A C voi d u a n g tron bang tie'p goc A

Cty TNHH MTV DWH Khang Vtet

J,) Ggi A p B | , C ^ , D , Ian l u a t la trong tam cac tam giac BCD, C D A, D A B , ABC

Chiing m i n h rang cac d u o n g thang A A p BB,, C C p D D j dong quy tai diem G '

Huong dan gidi «st h /

N Taco: G A + GB + GC + G D = 2 G M + M A + MB + 2OT

= 2 ( G M + GP) + ( M A + MB) + (PC + PD) = 0

4^-"Omv

3 A A i = A B + A C + A D ; 4 A G = A B + A C + A D => A A j = - A G ^ ^ ^

=> A A j ; A G Cling p h u a n g hay A A i d i qua G ' ••'•'47}

T u a n g t u ta co BBi d i qua G; CCi d i qua G; D D i d i qua G '"^ ;

V a y t a c o A A , , B B j , C C i , D D , d o n g q u y t a i G tia ; T : > ,af' Bai 1.108: Cho t a m giac A B C co trong tam G, M la m o t diem t u y y Go!

A i , B j , C j Ian lugt la cac diem doi x i i n g v a i M qua cac trung diem I , J, K ciia cac canh BC, CA, AB C h u n g m i n h rang i K " ' a) Cac d u o n g thang A A ^ , BB^CCj dong quy tai trung diem O ciia m o i d u o n g

b) M , G, O thing hang va ^ = ^ ^ " ' ' ' '

M G 2 Jii <:r (, •

Huang dan gidi

a) Goi O la trung diem CCi f , j

A A j = A M + M A i = A M + M B + M C = A C + M B - !

2 A O = A C + A C j = A C + M B ( v i A C j B M h i n h b i n h hanh)

A A J = 2 A O hay O la trung diem A A i

T u a n g t u ta co BB^ = 2BO hay O la trung diem BBi Vay A A p BBj, C C j dong quy tai trung diem O ciia m o i d u a n g )I 6 b

2 M d = M A + M A 7 = M A + M B + J^^^

=> M , G, O t h i n g hang va ^ = | Sai 1.109: Cho tam giac ABC Goi M , N , P la cac tie'p diem ciia d u a n g tron noi

tie'p tam giac A B C v a i cac canh BC, C A , A B Goi A^ la d u ^ n g thSng di qua t r u n g diem P N va vuong goc v a i BC, Aj, la d u a n g t h i n g d i qua trung diem P M va v u o n g goc v a i AC, A^ la d u a n g thang d i qua trung diem M N

va vuong goc v a i AB Chung m i n h rang A^, Aj, va A^ dong quy