^JSJBm.
NGUYEN
TRONG
TUAN
(CHU
BIEN)
ThS.
DANG
PHUC
THANH
-
NGUYEN
TAN
SIENG
(Gido
vien cliuy§ii toan
va
nang
khieu)
hi,; 'Uit
Bdi
duong
hoc
sinh
gidi
nociffiT
Danh
cho hoc
sinh
kha,
gidi

Kien
thuc - Ky nang cua chuang
trinh,
tren ca so do tai
lieu
con
giiip
cho cac em tiep can,
giai
cac de thi Dai hoc, Cao dang va cac bai toan
nang cao danh cho cac hoc sinh kha -
gioi.
" •
Noi
dung cuon
sach
gom 3 chuang, tuan tu theo
sach
giao khoa
Hinli
hoc 10 hien hanh. Moi chuang gom cac chuyen de tuong ung vai cac bai
hoc (§) duoc
trinh
bay theo cau true sau
A.
Tom tat li thui/et. ' ' I
B.
Mot so dang toan.
C.
Luyen

boat,
sang
tao, doc lap a moi em.
De
hoc tot tai
lieu
nay, hoc sinh can nSm vung li thuyet, cac dang
toan
va cac bai tap c6 lai
giai
mau. Voi
tinh
than tu hoc mot
each
nghiem
tue, khoa hoc,
chiing
toi hi vong rang tai
lieu
nay c6 the
giiip
cho cac em cai
thien
dugc nang luc hoc toan cua ban than va vuan len
trinh
do kha,
gioi.
Dii
da
CO

Vu Van Hoa Khang
Viet.
71,
Dinh
Tien
Hoang, P. Dakao. Quan 1, TP. HCM
Tel:
(08) 39115694 - 39111969 - 3911968 - 39105797 - Fax: (08) 39110880
Hoac
Email:
[email protected]
Cty
TNHU
MIV
DVVH
Khang
Vi£>
Chirofng
1
VECTO
§1.
KHAI
NIEM
VECTQ
A. TOM TAT LI
THUYET
m
1.
^inhnghia '
•

•
Gia
ciia
vecto
AB:
Cho AB khac 0.
Duong
thJing
AB
dugc ggi la gia
ciia
AB.
•
Hai vecto cung phuang: Hai vecta
dugc ggi la cung phuang neu
chiing
CO
gia song song
hoac
trung nhau.
•
Neu hai vecta cung phuang thi
hoac
chiing
cung huong,
hoac
chiing
ngugc huang.
Chu
y. Vecta - khong AA c6 gia la mgi duang thJing qua A; 0

ki hi^u la
I
AB
I.
Hai
vecta a va b ggi la bang nhau neu
chiing
cung huang va cung dg dai.
ta
viet
a = b.
Boi
liuchig
IISG
Hinh hoc 10
B.
MOTSODANGTOAN
i)ang
1.
So
sdnh
cdc
vecta
Sit
dung
cdc
dinh
nghta
vehai
vecta

Cling
huong.
b) Dung.
c) Dung vi neu AB = "BC thi
AB,BC
cung huong va AB
=
AC. Suy
ra
A,B,C
th^ng
hang
theo
thu tu do va AB = AC. Vay
B la
trung diem doan AC.
d) Sai vi voi A,
B
phan biet thi cac
vecto
nay
nguoc
huong.
<Bdi
2.
Cho tam
giac
ABC. Goi M,
N
Ian luot la trung diem ciia AB va AC. Hay

va
AC khong cung phuong;
=
2
MN
•Z
b) Hai
vecto
BC
va
MN cung huong va
BC
c) Hai
vecto
AN
va
NC bang nhau;
B
d) Hai
vecto
MA
va
MB cung phuong,
ngugc
huong va ciing do dai;
e) Hai
vecto
BC
va
NM ciing phuong,

sao cho:
a)AM
= MB;
^ ^
b)AB = BM.
Giai.
a)
M
la trung diem doan AB.
>
b)
M
la diem doi
xiing
ciia
A
qua B.
,v '
117 r,>
' Cty TNHH
MTV
DWH
Khmig
Vir.
<Bdi
4.Cho
tam
giac
ABC. Hay xac
dinh

E
la
die'm tren
tia
CA
sao
cho CA = AE.
(Hinh
ve). Si •••
C.
LUYEN
TAP ^^'^^^ ']
' •
1.1. Cho hai diem A,
B
phan biet. Hay
so
sanh
phuong, huong, do dai ciia hai
vecto
AB
va BA.
'
' • \
d&n
giai
Hai
vecto
AB
va BA

huong Ngugc huong
Bang
nhau
AB
va BC
MN
va BD
AB
va
CD
MA
va BC
MB
va
DA
DMva
BA
BA
va DC
J-Iaang
dan
giai
a)
Vecto
AB
ngugc
huong voi
vecto
DE
.

Vecto
FE
cung huong voi
vecto
BC
. v/
g)
Vecto
BD
bang
vecto
DC.
. . ; . -
h)
Vecto
EC khong cung phuong vgi
vecto
BF.
4^^,,
:,,
Boi
tluimg
HSG
Hinh
hoc 10
1.3. Cho ba diem A, B, C. Co nhan xet gi ve ba diem do neu
:
a)
AB=BC
; b)

v
1.5. Cho ba diem A, B, C phan biet. Ket luan gi ve ba diem A, B, C neu
:
a)
Vecto
AB cung phuong voi
vecto
BC;
b)
Vecto
AB cung huong voi
vecto
AC
;
c)
Vecto
AB
bang
vecto
AC
.
Jiicang
dan
gidi
a) Ba diem A, B, C thang hang.
b) Ba diem A, B, C thang hang va B, C nam ve cung mot phia doi voi diem A.
c) Hai diem B, C trung nhau.
1.6. Cho hinh binh hanh ABCD. Goi E,
F
Ian luot la trung diem cac

Mat
khac,
AADG = ACBH => AG = CH.
Nhu
the AGCH la hinh binh hanh. Tu do AH = GC.
Chu y:ABCD la
hinh
binh
hanh
<^A,B,C
khong
thang
hang
va AB = DC
CUj
TNHHMIV DVVII
Khnug
Vict
1.7. Cho
tijf
giac
loi ABCD. Co ket luan gi ve tu
giac
ABCD neu
AB = DC va
AB
AD
Jilzang
ddn
gidi

AB-a, BC = b
.
Khi
do
vecto
AC
duoc
goi la tong ciia hai
vecto
a
va
b.
Ki
hieu AC =
a + b.
2. Tinh
chat
• a + b = b +
a; *a
+ 0 =
a; *
(a + b) + c = a +
3. Cac quy tdc
• Quy tac ba diem
: •
Voi
ba diem A, B, C tuy y ta luon c6 AB + BC = AC
.
• Quy tac hinh binh hanh
:

qui tac ba
diem
Va qui tac
hinh
binh
hanh.
5Sdt
5. Cho 4 diem A, B, C, D tuy y. Chung minh rang
AB + CD = AD + CB.
7
Boi
dudng
HSG
Hinh
hoc 10
Giai
Theo
qui tac ba diem, ta c6: AB + CD = AD + DB + CB + BD
Ap
dung
tinli
chat
giao
hoan
ciia
phep
cpng, ta c6 :
AB + CD = AD + CB + DB + BD = AD + CB.
<8di 6. Cho tii
giac

chat
cua
tong
hai
vecto,
ta rut gon
kei qua
phep
todn
tong
cdc
vecto.
• Khi
ttnh
do dai
vecto
ta
thuang
xem do dai do la
canh
cua mgt
tarn
gidc
ndo do.
^di 7. Cho tam
giac
ABC. Xac djnh cac
vecto
u
= AB + CA, v = AB + CA + BC.

TNHH
MTV DWH
Khang
Vil-t
•
Theo
qui tac hinh binh hanh: OB + BD = OD:
• Dung hinh binh hanh
BACE.
Khi do
OB + BD OD =OD =
.72
AB + AC = AE:
AB + AC AE
= AE.
Ta
CO
tam
giac
ADE vuong tai D
biet
AD = a, DE = 2a.
Vay AE^ = AD^+DE^ =5a^ => AE =
a>y5.
Nhuvay AB + AC
=aV5.
' ' ' '
'bang
3.
Chung

man
dang
thuc
ve
tong
cdc
vecto
^di 10. Cho doan thang AB. Tim tap hop cac diem M
thoa
man
dSng
thuc
:
V
^ IMA + MBI=IABI.
Giai.
Ap
dung qui tac ba diem voi O la trung diem AB, ta c6 ^'
MA
+ MB=MO + OA + MO + OB = MO + MO + OA
Dvmg
vecto
OC = MO, ta c6
MC = Md + OC = MO + MO (hinhve).
M
O
Khido IMA + MBI=IABI o
IMO+MOI-IABI»IMCI=IABI
<=>
MC = AB 2MO = 20B o MO = OB.

phuong.
b) Diing. Neu I la trung diem
doan
MN thi MI = IN, do do:
MI
+ NI = IN + NI = 0.
c) Diing. Ap dung qui tac ba diem ta c6:
AB = CD AC + CB = CB + BD <::> AC = BD.
1.9. Su dung qui tac ba diem de
riit
gon cac tong
vecta
sau day:
a) u = AB + CD + BC + EA; b) v = AB + BC + CD + FE + DF.
Jiu&ng
ddn
gidi
a) U = AB + CD + BC + E^A = (EA + AB) + (BC + CD) = EB + BD = ED.
b) V = AB + BC + CD + FE + DF = AC + CD + DF + FE = AD + DE = AE.
1.10. Cho tam
giac
ABC
deu
canh
a.
a) Xac djnh va tinh do dai cac
vecto
u = AB + AC , v = CA + BA.
b) Goi M, N Ian lugt la trung diem
ciia

Khnitg
Viet
1.12. Cho 4 diem A, B, C, D. Chung minh rang AB = CD khi va chi khi trung
diem
ciia
hai
doan
thang
AD va BC trung nhau. -
•\ \'.i,
JIu&ng
ddn
gidi
,r„
Goi I va J Ian luot la trung diem cua AD va BC. Khi do ,j /j . ^
I AI = iD, JB = CJ. I
. Ta CO AB = CD AI + IJ + JB = Cj + Jl + ID » IJ = fl « IJ = 0
<=>
I = J.
1.13. Cho hinh
chCr
nhat
ABCD
voi AB = 2a, AD = a .
Tinh
AB + AD
AC + AD
AC + BD
AB + AD
= 2a.

do cac tu
giac
ABDC
va AMDN la hinh
binh
hanh.
Do do
theo
qui tac hinh binh
hanh
AB + AC = AM + AN = AD. D,
1.15. Cho tam
giac
ABC. Goi D, E va F Ian lugt la trung diem
ciia
cac
canh
BC,
CAva AB. Chung minh: AD + BE+ CF-d. : I.T
Jiuang
ddn
gidi
Cdch
i;Ta c6 :
AD
+ BE + CF =AB + BD + BC + CE + CA + AF
= (AB + BC + CA) + (BD + CE + AF).
Dira vao tinh
chat
duong trung binh trong

Do
do
AD
+
BE +
CF =
AE + AF +BF + BD
+
CD +
CE
=
AF+BF
+
BD+CD
+
CE+AE.
s^ n
Vi
D, E va F Ian
krot
la trung diem
ciia
cac canh BC, CA va AB nen:
AF + BF
=
0,BD +
CD = 0 va
AE + CE
= 0. 3"^^'
Vay AD + BE + CF=0.

gidi
a) Theo qui tac
hinh binh
hanh, ta c6
BB'
+
CC' +
BA +
CA
linn
=
BB'
+
BA
+ CC +
CA
=
BA'
+
CA'.
b)
Taco
BA' = BA
+
AA',
CB' =
CB
+
BB^,AC"
=

ciing
phuong voi
vecto OD.
b)
Chung
minh
rang OA
+ OB +
OC
+ OD +
OE-0. v.*^,,*. '
,u/^i;.
Jiuang
ddn
gidi
a) Truoc he't
chii
y rang do
tinh
chat doi
xiing
cua ngu giac deu nen OD 1
AB.
Dung
hinh binh
hanh
AOBF.
Khi
do
li

+ OB
+ OC + OD
+
OE
cimg
phuong voi OB
(vi
vai tro hai vecto OB, OD nhu nhau)
Vay
chi c6 the tong cac vecto tren la vecto - khong .
1.18. Cho hai vecto a va b khac vecto 0 .
a) Chung
minh
rang
la+bl<lal
+
lbl;
dau " = " xay ra khi nao ?
i;
ii
b)
Ap dung : Tim tap hop cac diem M thoa man dieu
kien
I
AM
+
MB 1=I AM
I
+1 MB
I,

. a + b _ ,
AC
=
AB
+ BC khi AB va AC
cijng
huong.
Vay
la +
bl<lal
+
lbl;
dau " = "xay ra khi a va b
cijng
huong. ; . '
b)
Dua vao ket qua cau a) ta thay rang
IAM
+
MBI=IAMI
+
IMBI
^
I
AB
1
=
1
AM
I

Vecto doi cua vecto a (ki hieu -a) la mot vecto ngugc huong voi vecto a
va
CO
cimg
do dai voi
vecto
a. , <!• 4' , ^
, 'u
•
. id V • OA
•-• i;
{j-y-jv
•Mv:uinm
:>(./.
Jiieu
cua hai vecto:
a)
Djnh
nghia:
Hieu
cua hai vecto a va b, ki hi|u a-b la tong
ciia
vecto a
voi
vecto
doi cua vecta b.
vj.u,» ji.
.
a-b = a
+

hinh
hanh.
\
<^di n. Cho
hinh
binh hanh
ABCD
va
M
la diem
tijy
y. Chung minh rang
MA-MB = MD-MC.
Giai.
^
.
: , :
Do
ABCD
la
hinh
binh hanh nen BA = CD.
(1)
Chuyrang
M
= MA-MB va CD = MD-MC.
(2)
Tu
(1) va (2) ta
CO

tinh dp ddi vecta ta thuang xem dg ddi do Id canh ciia
mot
tam giac
ndo
do.
<^dil3.
Cho tam giac ABC.
a) Hay xac dinh cac
vecta
u = AB
-
AC,
v
= BA
-
BC
-
CA.
b) Xac djnh diem
M
sao cho MA + MC
-
MB = 0.
c) Xac dinh diem
N
sao cho AN = AB + AC
-
CB.
a)
Taco

la
hinh
binh hanh. Nhu the AB + AC = AD.
14
Cty TNHH
MTV DWH
Khang Viet
Vay AN = AB +
AC-CB
= AD + BC = AD + AM
Vay
N
la dinh thu tu cua
hinh
binh hanh ADNM.
Cac diem M,
N
dugc xac dinh nhu tren
hinh
ve.
« »
^di
14. Cho
hinh
thoi
ABCD
c6 tam O, AB = a va ABC = 60".
Xac dinh va
tinh
dp dai cac

, , •
Vi
ABCD
la
hinh
thoi
CO
AB = a va
ABC = 60° nen BD = 2BO = a x/3 (vi
AABC
la tam
giac
deu).
Vay I
AB-AD
I
=DB = a>y3.
©ang-
3. Chitng minh tinh chat hinh hoc
^di 15. Cho tu
giac
ABCD
thoa man dong thoi cac dieu kien sau day
:
ii)
i) AB = DC;
ii)
AB + AD
Chung minh rang tu
giac

a va b
khong cimg phuong. Chung minh rang
:
Ne'u I
a - b I =
I
a + b I
thi
vecta
a
va
b
c6 gia vuong goc vai nhau.
Giai,
Vai hai
vecta
a va b
khong cung phuang, tu diem
O
bat ki, ta ve OA = a,
OB = b. Dyng
hinh
binh hanh
OACB,
khi do
BA =
a-bvaOC
= a+b.
Dodo
la - b I = la + b

IBA
+ MBI=iCM-CBI. , . , . , n
• • ' Giai
,<
j ^ , '/^ -tip
Ta c6: IBA + MB
1=1
CM - CB 1^1 BA - BM
1=1
BMI
• "'
<=>
I
MA I=IBMI <::> MA = MB.
Vay tap hop cac diem M la duang trung tryc doan thang AB.
C.LUYENTAP
1,19,
Chon khang
dinh
diing
trong cac khang
dinh
sau
a) Hai
vecto
doi nhau c6 dO dai bang nhau.
b) Neu hai
vecto
AB va AC doi nhau thi A = C. |
c) Hai

giac
ABC va diem M
tiiy
y. Cac khang
dinh
sau day dung hay
sai?
a)MA-MB
= BA > b) BA-CM = AB-MC
c) MA-BA = MC-BC d) MA + MB = CA + CB.
J /
iiiij^
Jiuong
ddn
gidi
i
a) Khang djnh
diing.
b) Khang
dinh
sai vi '
BA - CM = AB - MC o BA + BA = CM + CM » BA = CM (sai).
c) Khang
dinh
diing
vi CA = MA - MC = BA - BC MA - BA = MC - BC.
d) Khang
dinh
sai. j « ,
• •

ti'nh
chat
ciia hieu hai
vecto
ta c6:
KF = AF -
AK,
EH = BH - BE, GJ = CJ - CG,
IL
= DL - DI
Tir
do
KF +EH + Gj +
IL
= AF-Ak>M-BE+
CJ-CG
+
DL-Di
= (AF - BE)
+
(BH - CG)
+
(C|
- Dl) +
(DL
- AK) = 0 '
b) Ta CO
EL = EF + FK + KL = BA + FK + AD, HI = HB + BA + AD + DI.
=> EL - HI = FK - HB - Di = FK + CG - CJ = FK + JG = FK - Gj.
1.23.

BA-BM
=
CB-CM
MB
« MA = MB
<=>
M nam tren duang trung true doan
AB.
Gia sii M thoa man
BA
-
BM
=
IBA
-
BC
MA
CA
<=>MA
= CA
<=> M nam tren dirniiv
lirin
hi
in A,
]}j\f\
17
Boi
tiuaiig
use,
lliitli

+
DA
= DC
=
2a.
Dung
hinh binh hanh DACE.
Khido
AD + AC = AE=> AD + AC
=|XE|
AD
+ AC =
N/ABVBE^
=
2aV2.
BC+BA-BD=BC+DA=BC-BC=0
BC+BA-BD
=
0.
1.25.
Cho
hinh thang vuong ABCD
c6 hai day AB = a, CD = 2a,
duong
cao
AD
=
a.
Hay
xac

a.
AC-BD =
BF-BD
=
DF
=>
AC-BD
= DF
=DF =
3a.
DA-AB-CD
= DA + BA-CD
= EA-CD =
CB-CD
= DB
DA-AB-CD
= DB
=aV2.
AB-EA=AB+AE=AC
=>
AB-EA = AC
=asl5.
1
N
D
c
F
AC
-
DA

AM
+ BN + CP = 0r:^AB +
m
+
BC
+ CN + CA +
AP
=
6
hay
(AB + BC + C\ +
(AP
+
BM
+
CN)
= 0 <^ AP +
BM + CN =
0,
vi
AB
+
BC + CA =
0.
18
CUj
TNHII
Ml/V
DVVII
KItiuig

=
^(^Liy±^=2.
(2)
X
y z
x +
y
+
z
.•,{;,.
,
•
.
(2) ^ l + ^ = l + - = l + - = 2 x = y = z.
vyA;r,
>/,;:;
;
X
y z
VayAABCdeu. <*k'if:/
^ '
§4. PHEP NHAN VECTQ VQI MOT
SO
A.
TOMTATLITHUYET
1.
'f)inh nghla r 7
Tich
cua
vecto

ka
nguoc
huong voi
vecto
a
2) Do dai ciia
vecto
ka
bang
I
k
I.
I
a I.
2. 'Tinh chat
• Voi moi
vecto
a, b
va mpi
so
thuc k,
1
ta c6 :
i.r-'
1) k(la) = (kl)a;
^
2) (k + \)a = ka + la
;
3) k(a + b) = ka + kb;
k(a-b)

+
MC
= 3MG.
3.
^ieu kien de hai vecta ciing phuang
•
b
Cling phuong
a (a ;t 0) o 3ke
M:
b =ka.
•
Ba
diem phan biet
A,
B, C
thang hang
o 3 ke K :
AB = kAC.
4. S jeu thi mdt vecta qua hai vecta khong cung phuang • '
Cho hai
vecto
khong cijng phuong
a va b.
Khi
do
mpi
vecto
c
deu

can xdc
dinh
nen la
diem
ciioi
vecta.
<Bdi
18. Cho tarn
giac
ABC. Hay xac
dinh
cac diem D, E, F sao cho :
a) AD = 2AB ;
b) AE = ^AC ;
Giai.
c) AF = Ad + AE.
a) Ta CO AD = 2AB
<=>
AD
cung huong voi AB va AD =2AB
b)
Taco
AE = ACc:>AE
2
nguoc
huong voi AC va AE = ^AC.
c) Dung hinh binh hanh ADFE. Diem F la diem duoc xac djnh (nhu hinh ve).
^ang2.Xdc
dinh
diem

• *
•
- Su
dung
hieu
cua hai
vecta
ciing
diem
ddu MA - MB = BA
^dj
2
9. Cho tam
giac
ABC. Xac djnh cac diem M, N thoa man ding thuc sau :
a) MB
+
2MC = 0;
a)
Taco
b)NA
+ NB + 2NC = 0.
Giai.
MB
+
2MC = 0 MB = -2MC
<=>
MB, MC
nguoc
huong va MB = 2MC

qui tac ba
diem,
qui tac
hinh hinh
hanh
debieh
dot
vecta.
• Nen
chon
hai
vecta
ca so
(khong
cung
phuong).
(Bdi
20. Cho tam
giac
ABC. Cac diem M, N tren canh AB va P, Q tren canh AC
sao cho AM = MN = NB va AP = PQ = QB.
a) Tinh MP, QN
theo
vecto
BC .
b) Tinh cac
vecta
MQ, NP
theo
cac

NP = NM + MP = IBC -
i
AB
= i(AC - AB) - -
AB
= AB +
i
A^^^
3 3 3^ '3 3 3
Nhu
the MQ + NP = AC - AB = BC.
^di21. Cho hinh binh hanh ABCD. Tren duong
cheo
BD lay cac diem G va H
sao cho DG = GH = HB.
a) Chung minh rang AB + AD = AG + AH .
. o
b) AH cat BC tai M, AG cat DC tai N. Chung minh rang AM + AN = - AC .
Giai.
a)
Taco
AG = AD + DG ;
AH=AB+BH;
Boi liuihig HSG Hinh hoc 10
suy
ra:
AG
+ AH = AD + AB +
DC
+ BH

ma AM la duong trung tuyen
nen
AM =
-AH.
' " '*
,'•>,•
\
2
Chung
minh
tuong tu ta
ciing
c6 N la trong tam cua
AADC,
ma AM la
3
duong
trung tuyen nen AN =
—
AG
.
Dodo:
AM + AN =
|(AG
+ AH) =
-AC.
^ang 4. Chung minh ha diem thang
hang.
•
De'chiing minh ba diem phdn biet A,B,C thang hang ta chung minh hai

GQI
J la trung diem doan AB. Khi do lA + IB = 21].
Do
do lA + IB + 2rc = 6 o 2
(ij
+
IC)
= d <=> I la trung diem doan JC.
b) Ap dung qui tac ba diem, ta c6
NW
= MA + MB + 2MC = 5^ +
IA
+
Ml
+
iB
+ 2(N«
'
= 4Mi + lA +
IB
+
2IC,
vi lA +
FB
+ 2rc = 0 nen
22
MN-4MI.
Tu
day
suy

B,
I, N thang hang.
Giai.
1
a) Ta
CO
AN =
-AC
=> AN =
-
AC
va BN =
kBC
'
4 4
Ta
CO
0;A
BI^^^^^^^BA.JJBC.
2
2 2
1
BN=BA+AN=BA+-AC
4
=
BA +
-(BC-BA)
= -BA +
iBC.
b) Ba diem B, I, N thang hang khi va chi khi BI, BN la hai vecto cung phuong.

tam
giac
ABC.
' , .
(Bai
tap Dai so'10 nang cao)
Giai.
a) Ggi G' la trong tam cua
AABC,
tac6:G'A + G'B +
G'C
= 0.
Dodo
GA
+
GB
+
GC-0
o
GG'
+
G'A
+
GG'
+
G'B
+
GG'
+
G'C

GC
=
30G +
GA
+
GB
+
GC
=>
GA
+
GB
+
GC
= 0.
Vay
G la trong tam
ciia
AABC.
23
Boi
iditnig
use Uinli hoc 10
jVhdn
xet. Tir lai ^iai cua hai
loan
nay ta c6 cdc kct qua sau :
G la
tning
tam cua

+ GB' + GC = GA + GB + GC + AA' + BB' + CC = AA' + BB' + CC.
Talaico
AA'=
2AB,BB'=
2BC,CC'=
2CA. , . ,; , .
Tir
do GA' + GB' + GC = 2(AB + BC + CA) = 2.0 = 0. ; '1
Suy ra G la trong tam cua tam
giac
A'B'C.
Vay hai tam
giac
c6 cung trong tam. """I J
^ang 6. Tap hap
diem.
• AM = kv v&i A
codinh
va v
khong
doi thi tap hap cdc
diem
M Id
duong
thdng
qua A vd
cung
phuattg
vai gid
vecto

bang
I
v I.
(Bdi
26. Cho tam
giac
ABC. Tim tap hop cac diem M sao cho
a) MA + kMB - kMC ;
b) MA + (l-k)MB + kMC = 0.
. Giai. "/V'
a) Gia su diem M
thoa
man dang thuc MA + kMB = kMC.
Khi
do ta c6: MA = k(MC-MB) = kBC
Suy ra hai
vecto
MA, BC cung phuong. Vi
vecto
BC c6' djnh nen tap hop
M
la duong thang qua A va
song song
voi BC.
b) Gia su M la diem
thoa
man MA + (1 - k)MB + kMC = 0. Ta c6
OA
Cty TNHH MTV DWH
Khaug

,
Gia su M la diem
thoa
man dang thuc da cho, ta c6 /
MA
+ MB
MC + MD
<i=>
2MI
2Mj « MI = MJ
Vay tap hop cac diem M la duong trung true doan IJ.
b)
Taco
2MA-MB-MC = MA-MB + MA-MC = BA + CA .
Nhu
the
vecto
2MA - MB - MC la
vecta
khong doi. Do do dg dai cua no la
l2MA-MB-MCI=a khongdoi.
Goi K la diem
duoc
xac djnh boi KC + 2KD = 0 . Khi do K co djnh.
Gia su M la diem
thoa
man dang thuc da cho, ta c6 "
o a =
2MA-MB-MC
<::>l3MKI=a«MK

tlumtg
USG Ilhih hoc 10
Cty TNHH MTV DWH KItaiig ViSt
1.28. Cho a
khac
0.
Cac khang
dinh
sau day dung hay
sai?
a) Hai
vecto
a
va-3a
ciinghuang;
b)
Vecta-3a
c6 do dai gap ba Ian do dai
vecto
a; ^ ,
3 -
c) Cac
vecto
-—a
va5a
cung phtrong ;
d) Cac
vecto
2
a

-0A+20B
2
V57
-a.
I'H) .DHA.
"ifii;^
r.';>:.f
t'>.] j
• Chmig minh he
thiic
vecto
1.31. Cho tu
giac
ABCD c6
I,
J la trung diem cua hai duong
cheo
AC, BD.
Chung minh rang
a) AB
+
CD = 2rj; b) AB + AD + CB + CD = 41].
Jiimng
dan
gidi
a) AM = kAB ;
b) MA = kAB ; c) MB = kAM ;
d) BM = kBA; e) AB =
kAM
;

g) BA =
—AM.
6; 2
AB+AD+CB+CD
= 2 (AJ
+
Q) = -2 (JA + JC) = -4ji = 4IJ.
1.30. Cho tarn
giac
OAB vuong tai O voi AB = a va A = 60". Hay dung cac
vecto
sau day va
tinh
do dai cua chiing:
1^
3^^.
OA
+ 20B, - OA +-OB, -
OA
+ 20B.
2 2
Jiuang
ddn
gidi
Truoc
het ta c6 OA - -, OB = —
2 2
• Dung diem K
thoa
OK = 20B.

=
-OA,ON
= -OB.
2
26
60"
a
-I
N
Q
MN
= MA + AD + DN va MN = MB + BC + CN
Tudo
2MN
=
(MA
+
MB)
+
(DN
+
CN)
+
AD
+ BC
Chii
y rang: MA + MB = DN + CN = 6 •'
Tu
do suy ra dang thuc can chung minh .
b) Do M, N la trung diem cua AB, CD

1-k
1.34. Cho tam
giac
ABC. Goi M, N Ian
lirgt
la hai diem tren hai
canh
AB, AC
sao cho AM = 2MB va 2AN = 3NC. Goi I la trung diem doan MN. '
1
^ 3
a) Chune minh AI = —AB + —AC.
^ ^ 3 10
b)
GQ'I
K la trung diem
canh
BC.
Tinh IK
theo
cac
vecto
AB,
AC
.
Jiuang
ddn
gidi
a)
Truac

Jiuang
dan
gidi
IA-IB
+ 2IC =
0<»BA
+ 2IC = 0
^ 1
1
<=>IC
=
—BA«CI
= -BA.
2 2
I
la diem dugc xac djnh nhu hinh ve.
b) Goi D la trung diem AB .
Khido
JA + JB = 2JD.
Tudo jA + jB + 3jC =
0<=>2jD
+ 3jC = 0. gf-
Nhu
the J la diem nam trong doan CD sao cho 2JD = 3JC (hinh ve).
c) Ta
CO
2KA + KB - KC = CA o 2KA + CB = CA o 2KA = CA - CB
1
2KA = BA o AK = - AB <=> K la trung diem doan AB.
Cty TNHH MTV DWH

Jiuang
ddn
gidi
a) De tha'y I la diem doi xung ciia A qua B
va J la diem nam tren doan AC sao cho
3JA = 2JC.
Taco
ij =
A|-AI
= -2AB + -AC. (1)
5
Goi M la trung diem ciia BC. Khi do
AG
= -AM = -
3 3
AB + AC
=-AB+iAC
3 3
1
1
-5-
1
Vay IG =
AG-Ai
= -AB + -AC-2AB = —AB + -AC. (2)
3 3 3 3
b) (1) va (2) ^ Ta c6 5lj = 6IG = -lOAB + 2
AC,
tu do ta c6
I,

va G^ + G2B + G2C =
3G2G7
Tu
do suy ra dieu phai chung minh .
29
Boi ditaiig MSG Hinh hqc 10
1.38. Cho tarn
giac
ABC. G(?i D la diem tren canh BC sao cho DC=3DB, E Ici
,v
diem tren tia doi ciia tia BA sao cho AB = 2BE. Dat AB = a, AC = b.
a)
Tinh
cac
vecto
AD, DE
theo
cac
vecto
a, b.
b) Goi I la trung diem AC. Chung minh rang D, E, I thing hang va D la trung
diem EI.
AP-AB-AC
c) Xac dinh diem N tren mat phang sao cho SNA + 2NB + 5NC = 0.
d) Tim tap hop nhirng diem P tren mat phang sao cho
k>0 la so cho
truoc.
Jiuang
ddn
gidi

lONI.
. Dodo 5NA + 2NB + 5NC = 0c5 2NB + 10NI =
0c>NB
+ 5NI = 0
Vay
N la diem tren doan BI sao cho NB = 5NI.
d) Goi E la dinh cua hinh binh hanh
ABEC.
Khi do AB + AC = AE
=
ko
AM
-
AB-AC
-ko
AP-AE
EP
=
k o EP = k
Vay
tap hop diem P la duong
tron
tam E ban kinh k.
1.39. Cho tu
giac
ABCD
va diem M tuy y. Chiing minh rang cac
vecto
sau
khong phu

UVVii
Killing
1,40. Cho hinh binh hanh
ABCD
c6 tam O. Goi M, N Ian luat la trung diem
ciia
cac canh BC va CD. Dat AB = a, AD = b.
a)
Tinh
cac
vecto
BN, AM
theo
a, b.
b) Goi I la
giao
diem ciia BN va DM. Chung minh rang: IB + IC + ID = 0 .
3r
c) Ggi K la diem xac djnh boi AK = - h
•
Chung minh N la trung diem doan MK.
Jiuang
dan
gidi
- Ir
a) AM = a + -b, BN = a + b
b) I la
trong
tam cua
ABCD.

Tiep
theo
dung hinh binh hanh
ACMD.
Khi
do AM = AC + 2AB . M la diem can xac dinh.
b) Ta CO
AN
= AB + BN=AB + kBC
- AB + k(AC - AB)= (1 - k)AB + kAC
A,
M, N thang hang <=> AM va AN ciing phuong
ol^
=
-<=>l-k
= 2kok = i.
2 1 3
1.42. a) Cho hai
vecto
khong ciing phuong a, b . Chung minh :
ma + nb = 0 <=> m = n = 0.
33
Boi
dumig
HSG
Hinh
hoc 10
b) Cho tam
giac
ABC c6 cac

-a).GC
= 6
Vi
GB va GC la hai
vecto
khong cung phuong nen
theo
cau a), ta c6 :
b-a = c- a =
0<=>a
= b = c.
Vay tam
giac
ABC la tam
giac
deu.
• Tap hgp diem
thoa
man thiic
vecta
,
1.43. Cho tam
giac
ABC. Tim tap hop nhung diem M sao cho
a) MA = MB + kMC; b) MA + kMB + MC = 0 ;
b) (l-k)MA + (l + k)MB + MC = 0; d) MA + MB + 2MC = kBC.
' ' "
Jiuang
ddn
gidi

Khang
Vu
c) Gia su M la diem
thoa
man gia thiet bai toan, ta c6
(1 - k)MA + (1 + k)MB + MC = 0<::>MA + MB + MC + k(MB - MA) = 0
<:}.3MG
+ kBA = 0<::j>MG = —BA.
3 ^,
Voi
G la trpng tam tam
giac
ABC.
Vay tap hgp cac diem M la duong thang qua G va
song song
voi AB.
d) Gpi I la trung diem AB va J la trung diem CJ. Khi do
is^ + MB + 2MC - kBC o 2MI + 2MC = kBC o 4MJ
Vay tap hgp cac diem M la duong thang qua J va
song song
voi BC.
1.44. Cho hai diem A, B co'dinh. Tim tap hgp nhimg diem M sao cho
2^ ^ b) 2MA + 3MB = 4MA + MB
a)
c)
MA
+ 3MB =-MA-MB
3MA + MB-2MC
3MB-2MA-MC
Jlit&ng

Vay tap hgp cac diem M la duong trung tryc doan IJ.
c)
Truoc
het chu y r^ng ' •. ^ "
3MB-2MA-MC
= 2(MB-MA) +
MB-MC
=2AB
+ CB = U
Ggi I la diem
thoa
man 3IA + IB - 2IC = 0. Khi do ta c6
3MA + MB-2MC = 2MI
Tu
do, gia thiet bai toan tro thanh
3MA + MB-2MC = 3MB-2MA-MC
2MI
u
«.MI =
V^y tap hgp cac diem M la duong tron tam I ban
kinh
bang
lui
33
Boi
ditmg HSG lihih hoc
10
§5.TRyC
TOA
Dp VA

va
dupe
xac
djnh
boi he thue AB =
AB.i
(T
la vecto don vi cua true).
*
H§ thuc
Sa-lo:
AB + BC = AC
voi
mpi
diem
A, B,
C
tren true Ox.
2.
Jie
true toa dd
•
Trong
he true toa dp Oxy
a
= (x;y)oa =
x.T+yj
M
0
y

phuong vai a^O
<=>
x'
= kx,y' = ky vai k E R
•
Neu
I
la trung
diem
doan AB thi Xj =
^^A±^;
=
^^ +
76
•
Neu G la trpng tam tam giac
ABC
thi
_
XA
+
XB
+
B.
MOTSODANGTOAN
_XA
+
XB
+
XC

-
3AM
+
BM
+
MC
=
0.
Giai.
a) Toa dp
ciia
vecto AB la
2 -
(-3) = 5.
To?i
dp cua vecto BC la
5 - 2
= 3.
,
;
>
Toa
dp cua vecto CA la
-3 -5
= -8.
Toa
dp cua vecta -3AB + 4BC +
2CA
la -3.5 + 4.3 + 2(-8) = -19.
b)

X =-2 .
Vay
toa dp cua M la
X
=-2.
.
^dj
29.
Tren
mot true cho
4
diem
A, B, C, D.
Chung
minh
rang
AB.CD
+
AC.DB
+
AD.BC
=
0
(He thuc
Euler).
Giai,
Gpi
toa dp
eiia
4

khai
trien
va
riit
gpn ta eo
dieu
phai
chung
minh.
(I
i)Qng
2. Tinh todn
ve
toa do diem
-
vecto tren mat phang.
•
Sic dung cac cong
thUc
ve toa do diem
-
vecto.
•
Su dung dieu kien cua hai vecto ciing phuong.
.
•,{!.
:ai;;.|
-f
',r
j'iil;

+
Ta
CO
2a = (-6;
2),
3b = (12;
9).
-
' - ^ '
;
r
'
Tudo
2a-3b =
(-18;-7).
+
Ta
CO
-3a
= (9;-3), 4b = (16; 12), 2c = (-4; 12).
Nhu
the-3a + 4b + 2c =
(21;21).
t>)
Gia
su c = xa + yb.
-

30
'-3x + 4y =

?
; r^r
Bang
tqa dg chting minh hai vecta
AB,AC
khong cung phuang.
Suy ra A, B, C khong thang hang.
^di
Sl.Cho ba diem
A(5;-2),
B(l;3),
C(2;-4).
Chung
minh
rang ba diem A, B, C khong thang hang.
Giai.
Taco
AB-(-4;5),
AC-(-3;-2)
-4 5
5>t-—=>AB,
AC khong cung phuong. . |. ^, - .,.t
w
,
Vay
ba diem A, B, C khong thSng hang. lA' i ' >
' •
^di
32. Cho
A(-l;2),

1). Ta c6 '
•
AM
= (x +1
;-l),
AB
=
(4;-l).
•. r
>
at
>
V;>0(:H
rt>'(.;i>-JKX
*
A,
M, B thang hang
<=>
AM cung phuong voi AB
'inifo
stX (
«
^^ =
—<»x
+ l = 4
<::>x
= 3. Vay
M(3;l).
t)ang
4. Xdc dinh tga do diem M thod dim ki^n cho truac.

36
Cty
TNHH
MTV DWH Khang Vi$t
Giai.
a) Taco AB =
(2;6),
AC = (-1;7).
Ta
thay cac vecta AB va AC khong cung phuang nen ba diem A, B, C
khong
thang hang. Do do A, B, C la ba
dinh
aia mpt tam giac.
Gpi
G la trpng tam tam giac
ABC.
Khi do ta
CO
'
XA + XB + xc _ 4
XG-
5 -3
yA
+
ys+yc
_7
3
3
b)

-l
Vay
D(-2;-l).
Tam I cua
hinh binh
hanh la trung diem duong
cheo
AC. Do
/I
3^
do I
U'2
Bai
34. Cho tam giac ABC vai
B(l;-1),
C(6;-3).
Biet
trpng tam
ciia
tam giac nam
tren
true hoanh va trung diem M cua canh AB
each
deu hai tryc tpa dp. Xac
djnh
tpa dp diem A.
Gia
su A(a; b). Khi do trpng tam G
ciia
tam giac c6 tpa dp

"a + l = 3
=
-0 <=>
2
2
a + l = -3
a = 2
a = -4.
Vay
A(2;
4)
hoae
A(-4;
4).
V7
i
Boi
dumig
HSG
Hinh
hoc 10
CLUYENTAP
• Tpa dp diem,
vecta
tren mpt true > t
i^'
1.45. Tren true x'Ox cho hai diem A va B c6 toa do Ian luat la 2 va 8.
a) Tim toa do diem M biet rang MB + 2MA = 0 .
b) Tim toa do diem N biet rang NB-3NA-0. .,',<n i. f Mo
c) Tim toa dp diem P doi

Hay 50C = 30A + 20B OC = - OA + - OB (dpcm). t.,
5 5
.••' <
Gpi
I la diem thoa man 3IA + IB - 2iC = 0.
• Tinh
toan
ve tpa dp diem -
vecta
tren
mat
phang
1.47. Cho hai
vecta
a =
(2;5),b
= (2; 3). Khi do
vecto
v = 2a - 3b cung phuong
vdi
vecta
nao sau day ?. ,.| ,^
a) ^ = (!;!) b) xI^ =
(-4;-2)
c)
ii^
=
|V2; ^)
d)
Ii^

doi
nhau. 1
b) Neu
vecta
a 6 cung huang vdi
vecta
I thi no cd hoanh dp duong.
Cty TNHU MIV
DVVH
Khang
Vic
Neu
vecta
a 0 ngupc hudng vdi
vecto
j thi no cd tung dp am.
J) Neu
vecta
a khong cung phuong vdi ca hai
vecto
i va j thi ca hai toa do
cua no deu
khac
0.
e) Hai
vecto
a = (-3; V3) va b = (V3; -1) la hai
vecta
cung hudng.
Jiuang

xsao
cho hai
vecta
u,v ciing phuong.
b) Xae
dinh
x sao cho
vecto
2u -3v cung phuong vdi
vecto
i .
•
y,,^:^
•:r
• '
•
: Ji^g d^ri
gidi
,
,
^t^j
ejjp
yV-,f
a) Hai
vecto
u, v cung phuong khi va chi khi =
—
<z> x = -12
2
b) Ta cd 2u = -T +

Jiuang
dan
gidi
. \
a) Tim toa dp cac
vecto
2a + b = (- 8;]!), - a + 3b -2c =
(-2;5).'
"'
b)
Tacd
ka =
(-3k;4k),lb
=
(-l;31):^ka
+ lb =
(-3k-l;4k
+ 3l) . ^
^ . -3k-l + 2 = 0
Tacd: ka + lb + c = 0<=>-^ ^, „ o<i
4k + 31 = 0
5
1
= -^
5
Bdi
duang
HSG Hinh hpc 10
c) Ta thay hai
vecto

A(l;
2). j, - *•> > , - -v, .r (•H / n/ ? r.,
••^-vi^
'
h) Neu A(x;y) va CB=xi-yj vai y^O thi A va B doi
xiing
nhau qua true Oy.
c) Neu hai diem A va B c6 hoanh dp doi nhau, tung dp doi nhau thi trung
diem
doan AB la go'c toa dp O. i
d) Neu A(a; 1) va B(-a; 3) thi trung diem doan AB nam tren true hoanh.
JIudng
dan
gidi
a) Sai (phai sua lai la OH = 1, OK = 2,
b) Sai (doi
xiing
qua Ox), • ;'
^
- ^, '^^wVl,
c)
Diing,
d) Sai (nSm tren Oy).
^>no-nui:'fnh
v-,u
•
v:k>ll i,
• Xac dinh tpa dp diem
thoa
man dieu kifn cho

ABC. De c6 G
3 3 j
b) Lay D(x;0) la diem nam tren true hoanh. Khi do AD = (x + 2;-5)
Ba diem A, B, D thang hang
<=>
AB, AD cung phuang
X + 2 -"i 14
o = —<»3x + 6 =
20ox
= —. Vgy D
I
3
4n
Cty TNHH MTV DWH
Khang
Viet
Lay E(0; y) la diem nam tren true tung. Khi do CE = (-10; y - 4)
ABCE la hinh thang
<=>
AB,CE ciing phuang
,d£ =
^c>4y-16
= 30<=>y = —.Vay E
2 )
153. Trong mat phang
t<?a
dp cho ba diem A(2;-l),
B(x;2)
va C(-3;y).
a) Xae

-l.
••••
1 t ,
c) AB = (x-2;3),AC = (-5;y + l)
A,
B, C thang hang
<=>
AB,
AC cung phuang (x - 2)(y +1) +15 = 0.
1.54. Cho A(2;l),B(2;-l),C(x;-3),D(-2;y). ,
a) Tim x, y de ABCD la hinh binh hanh .
b) Tim
X,
y sao cho ba diem A, B, C tao thanh mpt tam
giac
nhan D lam trpng tam.
Huong
ddn
gidi
a)
Taco
AB = (0;-2),AC =
(x-2;-4),
DC = (x +
2;-3-y).
ABCD la hinh binh hanh <:>AB = DC vaA,B,C khong thang hang
'0.(-4)
+
2(x-2)7^0
x = -2

diem C nam tren tia Ox. Tim toa do B va C. ,
'
! JJu&ngddngidi ;
>
Goi C(xc ; 0) va B(0 ;
yB)
Ian
lirgt
nam tren cac truoOx va Oy sao cho AA\]Q
nhan
G lam trong tam. Ta
CO:
i , , , f
'
^
f3
+
0
+
xc=7^jXc=4
*
, , ,
|7 +
yB+0
= 9
[yB=2
I . ' ,
Vay
B(0 ; 2) va C(4 ; 0).
1.56. Trong mat

b)
Vol mpi diem M tuy y thi MA + MD = MB + ME = MC +
MF.,

Jiuang
ddn
gidi
a) ABCDEF la luc
giac
deu tam O. Khi do cac cap diem A va D, B va E, C va F
doi
xung qua tam O. Do do: OA + OD = OB + OE = OC + OF = 0 .
b)
Dedangc6
MA>MD
= MB + ME = MC + MF = 2K^
1.58. Cho tam
giac
ABC. Hai diem M va N Ian lugt di dong tren cac duong
thang AB va AC sao cho AM = mAB, AN = nAC . Tim dieu kien cua m va
n
sao cho vecto v = BN + CM cung phuong voi vecto BC . , . ^
Jiuang
ddn
gidi
Truoc het ta c6 BC =
-
AB
+ AC. :
Taco

:>
[Xr-
=2XK]
-x^ =-2-x
yc=2yN-YA=4-y
M
la trung diem BC
fxg
=2XM
-XC
=6 +
X
•J
,0
Tu
(1) va (2), ta c6: BC va v ciing phuong khi va chi khi
m-1 n-1
-1
1
•
<=>
m + n = 2.
•
C(-2-x;4-y).
lyB = 2yM-yc = -io + y
B(6
+ x;-10 + y).
P la trung diem AB
Xp
=

n
cua x va y sao cho ba diem M, N, B
thSnghang.
' 0,-'CTT 'ir!
4/A'b.
Jiit&ng
ddn
gidi
' ''^
a) Taco: AB-OB-OA
2y-10 = -4 [y = 3.
Vay
A(0;3),B(6;-7),C(-2;1).
= BD+iAC=-a b
2
2 2 2
BC=OC-OB=iAC+iBD=ia+ib
^)
Taco
AM = xAB = x
—a—b
.2 2 .
x
- X r
=-a b
2
2
AN
= yAD = yBC = y —a +—b
U

=
0
' '
22 22 ^ \ y
.'i*
Hay
xy-x^-2y
+
xy-2x
+ x^
=Ooxy
= x + y
Vay
dieu kien de M, N, C thang hang la xy = x + y. ' '
1.60.
Cho tam
giac
ABC . Tren cac canh BC, CA, AB Ian lugt lay cac diem M,
N,
P sao cho BM + CN
+
AP = 0. Chung minh rang cac tam
giac
ABC va
MNP CO
cung
trong
tam. • ,, .,,»,f >/r e/-v ^
<
O ui .cisi

CP= " 'f
, ^ \< I HI
Tu
do ta
CO
dieu phai chung minh. ' " " '* ' ' * ;
1.61.
Chung minh rMng dieu ki^n can va du de hai hinh binh hanh
ABCD
va
A'B'C'D'cocungtamla
AA'
+
BF'
+
CC'
+
DD'
= 0. , .,,
, , ,.,
JiuangMngidi
,,; ^ ^.^.^^.^
Gpi
I, r Ian lugt la tam ciia hai hinh binh hanh.
;
A
- ^1
(\
>:
: |/?

BB^
+
CC
+
DD'
= 4ir •
Tu
do suy ra dieu phai chung minh.
1.62.
Cho tam
giac
ABC.
M la diem tren canh BC sao cho BM =
SMC,
N la diem
doi xung ciia M qua C.
a)
Tinh
cac
vecto
AM, AN
theo
hai
vecto
AB = a, AC = b .
b) Gpi I la trung diem AM, J la diem tren AN sao cho AJ =
kAC.
Xac djnh k
debadi&nB, I,
Jthanghang.

4"" 4
^b) Ta
CO
t - 1
B
M 9
AMI
'iA^Mfx-'ilh
Ai
= -AM = -a + -b, AJ = kAN = a + —b, ;^«'i:;rfn-^,r? a
2 8 8 4 4
Bi
=
AI-AB
= ia + -b-a = —a + -b '
8 8 8 8
.,V:v^,:.,
BJ = AJ-AB = —a
+—b-i
= ^^a + —b
4 4 4 4 A"
0 Ba diem B, I, J thang hang khi va chi khi
BI, BJ
cung phuong
. -k-4 -7 5k 3 2(k
+
4) 10k „, „, , 3
hay
:
— = —

K
a) D^t AB = a,AD-b.Giasu — = z:^BN = zBI.
BI
' ^ ' .
'•
^'
Ti'nh
dugc
AM = ya + b,
AN
= (1 - z)a + —
b
45
Boi
diam^ liSC
Uiiili
hoc 10
Cac
vecto
AM, AN ciing phuong. Suy ra
, , 3 , 10
Tu
do vai V = — ta CO z = —
^ 5 13
1-z z/2
<=>z(y
+ 2) = 2
b) Goi O la tarn ciia
hinh
binh hanh. De dang tinh

. \>\k. .:l.: ,,Af''. k
a) Xacdinhdiemlsaocho IA + iB + iC + !D = 0.
b)
Chiing
minh OA + OB + OC + OD = 40I voi O la diem bat
ki.
•
c) Xac dinh diem M
deM
MA + MB + MC + MD
I
c6 gia trj be nhat.
d) Tim tap hop nhirng diem N sao cho AB cung phuong vai
vecto
v=NA+NB+NC+ND.
Jiwmg
dan gidi
a) Goi M, N Ian lugt la trung diem cua AB va CD.
Taco rA + iB =
2iM,iC
+ ID = 2iN
Nhuthe:
IA + IB +
IC
+ ID = 0^2IM + 2IN = 0
<=>
Ila trung diem doan MN.
b) Ap dung quy tac ve
hif
u

dat gia trj be nhat o MI = 0.
VayMs
Ithi
IMA + MB + MC +
MDI
dat gia tri be nhat.
d) Ap dung ke't qua cau b), ta c6: v = NA + r^ + NC + ND = 4NI.
Nhu
vay, v ciing phuang vai AB o NI ciing phuang voi AB <=> NI song
song vai AB.
Nhu
the tap hgp nhiing diem N sao cho v ciing phuang vai AB la duang
thang qua I va song song vai AB. ,w:
ft,,
fNlUl
MVV
DVVn
Khaug Viet
J
65- 8'^^ ABC. Cho d, d' la hai duong thang phan biet thay do!
nhung
luon song song vol BC. Biet d cat hai canh AB, AC Ian lugt tai M, N ;
d'
cat hai canh AB, AC Ian lugt tai P, Q.
Chung
minh
vecto
v = MQ + PN c6 huang khong doi.
Jiicang ddn giai r r
, AM AN , , AP AQ , ,. ^ , , ,

Nhu the '-iU
.••:\iV'X
i'^'^ v
.j ::,','
v
= MQ + PN =
lAC-kAB
+
kAC-lAB
' : ^ ''
=
(k + 1)AC -(k + 1)AB = (k + 1)(AC - AB) = (k +
I)BC.
'
-;4 , trung diem canh CD la
P(2;2).
Hay xac dinh tga do cac dinh A,
Vay
V luon cung huang vai BC.
1.66, Cho
hinh
binh hanh
ABCD
c6 trung diem AB la M(l;5), trung diem BC
laN
B,
C, D.
Jiuang ddn gidi
Cho
A(x; y). Vi M la trung diem AB nen

hoc 10
Tu
do ta CO
f7 + 2x ^ 3
2 ~2
^ 2
Vay toa do cac diem la A
x = -2
y
= 2
,C 5;- ,D
V 2y
1.67. Cho cac diem M(-2;3), N(0;4),
P(-4;l).
^ ;
a) Chiing minh M, N, P la ba dinh mpt tarn
giac.
, j
b) Gia sit M, N, P
thoa
man MA = -MB, NB = -2NC, PC = 2PA . ,
^
.
Hay xac dinh tpa dp cac diem A, B, C. ,. j , j ,
Jiucfngdangiai
vj
^
^^^^^ ^ ^ ,^
Q,,,,;;
^

XM=-
Vay hai diem can rim la A (4;
0),B(0;
6). nki :/
b)
Giasu
C(c;0),D(0;d)
Xet truong hgp MC = 2MD
Taco
MC =
(c-2;-3),MD
=
(-2;d-3)
c-2 = -4
-3 =
2(d-3)'
Nhuthe MC = 2MDo-^
c = -2
2
Vay hai diem can tim la
C(-2;0),D
0;-
\/
• Xet truong hpp MC = -2MD
f
9
Thuc
hien nhu tren ta
dxxgc
hai diem C(6; 0), D 0; -

vecto
AD =
(4;-l),
AN
=
Vi
hai
vecto
AN, AD cung phuang nen 4(-2) =
^21 ^
^ 5^
2a
2)
/ IS 21
(-1) =^ a = —
4
Tu
do ta tim
dugc
tam I —;0
V 4
-V
^9.
,B
-3
,B
—;-2
I2
J
,B

CO
gia tri be nhat.
M
chinh la
giao
diem ciia duong thang OG
voi
duong tron (O) ngoai tiep cua AABC.
Th^t vay, voi N bat
ki,
N ?t M, ta c6 :
OM=ON<OG+GN
<=>OG + GM<OG + GN,do do GM<GN
E5iern
M
dugc
xac djnh nhu tren hinh ve.
wA