A
+
AB AC < BC < AB + AC
AB BC < AC < AB + BC
TAM
B
C
AC BC < AB < AC + BC
A
+ nhau. TAM + nhau. + AB = AC
+
mang + - -
B
C
+
+ AB = AC = BC nhau.
A
TAM +
0.
nhau. 0.
+
C
B
0
C
0
TAM
1)2
2)2
3)2
+ + + BC2 = AB2 + AC2 +
A
+
B
C
0.
TAM + +
450
+ AB = AC + + + BC2 = AB2 + AC2
A
B
+ AM = BC :
1/ 2/ 3/
A
A
E
F
G
TRUNG
B
C
D
B
C
M
BM = MC = BC : 2 Trong
AG BG CG AD BE CF
2 3
ABC.
ABC
1/ 2/ 3/
A
A
K
L
H
B
C
H
CAO
C
B
I
d
Trong AH
ABC
ABC.
4/ 5/
A
B
H
C
H
B
C
A
AH
B
AB, AC
1/ 2/ 3/
A
A
d
O
TRUNG
d
C
C
M
B
B
MB = MC
Trong trung tr
: OA = OB = OC ABC
ABC.
2/ 3/ 1/
A
A
L
M
I
K
B
C
B
C
D
Trong ABC, ba Tia Oz
IK = IL = IM ABC.
ABC.
c
2
A
+ +
3
1
+
4
a
+
2
3
+ +
1 B
4
b
+ +
c
c
a
a
a
b
c
b
b
a
/ /
c
. a // b
b
/ /
c
a c
a a / / b c b c
b
c
/ /a b
/ /a b
Suy ra: + + +
B
A
. 1/
D
C
A
B
1/ AB // DC.
THANG . 2/
D
C
A
B
1/ AB // DC.
THANG . 2/
C
D
.
1/ AB // DC.
A
B
2/ AD = BC.
THANG . 3/
. 4/
D
C
5/ AC = BD.
1/
2/
A
B
3/
I
4/
C
D
5/
1/
A
B
. 2/ 3/
I
4/ AC = BD.
C
D
5/ IA = IC = ID = IB.
B
1/ 2/ AB = BC = CD = DA. 3/ 4/
THOI
A
C
I
5/ 6/ BD 7/
D
1/
A
B
2
1
2
2/ AB = BC = CD = DA. . 3/
1 450
4/ AC = BD.
I
5/ IA = IC = ID = IB.
6/ BD
1
2
2
1
7/
D
C
a
h
b
a
h
h
h
a
b
a
a
S
a h .
a h .
S
a b h .
Hthang
HBHS
1 2
1 2
a
d1
d1
a
d2
d2
b
2
S
S
a b .
Hthoi
d d . 1
2
d d . 1
2
HCNS
S
a
Hvuông
1 2
1 2
a/ a/
A
B
A
MA MB
MA MD NB NC
NA NB
N
M
M
N
ABC. ABCD.
B
C
C
D
b/ b/
A
B
A
ABC. thang ABCD. MN // BC
N
M
N
M
AB CD
1 2
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C
D
B
C
c/ thang ABCD, c/
A
B
A
MA MD
N
M
MA MB MN / / BC
M
N
MN
/ / AB/ / CD
C
D
C
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a/ b/
A
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A
B
I
I
IA = IB
b/ a/
d
C
D
d
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1 2
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MA MB
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A
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1
1
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I
2
2
:
1
2
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AC B D
D A AB C BC / / D
a/
b/
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MN // BC AM AN MN BC AC AB
A
B
C
MN // BC AMN ABC
A
x
ABC AE
A
C
B
M
C
E
B
D
AB DB EB AC DC EC
AM = BC : 2
A
k
ABC DEF theo ABC = DEF
h1
C
B
ABC
k
D
DEF
2
ABC
h2
k
h 1 h 2 Chuvi Chuvi S S
DEF
F
E
D
A
AB DE AC DF BC EF
(c.c.c) AB DE DF
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F
D
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(g.g) (g.c.g)
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E
F
4/
D
A
D
A
BC E F
AB DE
B
BC EF AB DE (ch-cgv)
C
E
E
B
F
C
F
A
D
D
A
B
(ch-gn)
C
E
B
E
C
F
F
cgv1
cgv2
cao
hc2
hc1
< tan ; cos < 1. <
ch
+ 0 < sin + CM: sin cot
trong
1/ sin =
+ = 900 cao trong 1/ (cgv1)2 = hc1 . ch (cgv2)2 = hc2 . ch 2/ cos =
2
2
2
1 cao
(
)
(
)
1 cgv 1
2
sin = cos 3/ tan = 4/ cos = sin 2/ cao2 = hc1 . hc2 3/ cao . ch = cgv1 . cgv2 1 cgv 4/ cot =
2
1
2
1
2
tan = cot
1 sin tan
sin tan
;cos ; cot
cos cot
1
2
1
2
cot = tan ch2 = (cgv1)2 + (cgv2)2
ch = hc1 + hc2
1/ tan
sin cos
300 450 600
3 /2
2 /2
2 / cot
1/2 sin
cos sin
3 /2
2 /2
1/2 cos
3
3 /3
4 / tan .cot
1
1 tan
3
3 /3
1 cot
cgv1
cgv2
) ) 3) cgv1 = cgv2
2) cgv = ch . 4) cgv1 = cgv2
ABC
(hay
Hay (O)
.
ABC
ABC
-
ABC
ABC -
ABC
-
A
m K trong
ABC
-
D
B
C
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K
C)
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-
MN
qua trung
OA
(O)
d
AB = AC
nhau
chung.
B
AB = AC
A
B
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A
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I
C
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cung
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x
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1 2
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M
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B
A
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C
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1 2
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1 2
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B
1 2
1 2
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2
1
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2
1
.
1
2
2
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1
C
D
B
(BCx
0
R, cung n0
R
A
R2
C =
R
R
S
O
n0
O
l
n0
O
S
l
2nR 360 lR 2
C = d
B
2R
S =
Rn 180
1/
a b
c d
a c b d
a d .
b c .
2 /
a b
c d
a b b
c d d
6/ A3 + B3 = (A + B)(A2 AB + B2) 7/ A3 B3 = (A B)(A2 + AB + B2) 1/ (A + B)2 = A2 + 2AB + B2 2/ (A B)2 = A2 2AB + B2 3/ A2 B2 = (A B)(A + B)
6*/ A3 + B3 = (A + B)3 3AB(A + B) 7*/ A3 B3 = (A B)3 + 3AB(A B) 4/ (A + B)3 = A3 + 3A2B + 3AB2 + B3 5/ (A B)3 = A3 3A2B + 3AB2 B3
c 1/ ax by a x b y ' ' c ' b b ' a a ' (d): y = ax + b (a 0)
c 2/ ax by a x b y ' ' c ' c c ' a a ' b b ' (d)
a c 3/ ax by a x b y ' ' c ' a a ' b b ' c c ' (d) -1
2 A
2/ 1/
xA )( )( xB
2
2
A A A B(x) 0
)(xA
A
A
AA .
A
A(x) 0
A B .
A B .
xA )( xB )(
B(x) > 0 0; B 0)
2
A
B
BA
B A
0 BhayA
B
B
0(
hayA
0)
A
B
2A B
A B
2/ ( A 0, B > 0) A B A B
A B
B
0
BA
2
BA
0)
A B
2 A B
A
0
A
B
0
( A 0)
A B
2 A B
B
0
( A < 0)
4/ 0 5/
A m. n .A
m n A
m A . A A .
n .
a
a
a a .
a
a
.
a
1
A A A A A . A a .a a 2 a 2:
m
B
a b b a
ab
a
b
. Am 2 A
B
A
B
2
2
3:
a b
a
b
a
b
a
b
m . Am B
2
BA A B
a b
2
ab
a
b
4:
3
3
a
1
a
3
3
1
a
a 1 a a a 1 a a 1 a a a b b a b a b a ab b
aa a
a
3
3
abba
ab
a
b
ab
a a b b
a
b
a
b a
ab b
a
b
a
b
a a b b a b a b a ab b
2/ 3/
2 + bx + c = 0 (a
1/ b: ax2 + c = 0 (a 0) 0):
ax2 + bx = 0 (a 0) x2 = c a x(ax + b) = 0
b
x 0 ax b
0
x 2
2
a
x
0
x
c a
b a
> 0 x2 > 0 b ; x 1 2 a
x 1
x 2
b a
2
0;
2 < 0
b a
= 0
1, x2
x 1
x 2
b a
< 0
. xx 1 2
c a
ax2 + bx + c = 0 (a
3/ 1/ 2/
x1
2:
1 + x2
a + b + c = 0 2 + bx + c = 0 (a a b + c = 0 2 + bx + c = 0 (a
1 . x2
c a
2 Sx + P = 0
x1 = 1, x2 = x1 = 1, x2 = c a
2
2
2
2
x 1
x 2
x x 1 2
1
2
4/ 5/ ax2
x x
3
3
3
3
x 1
x 2
x x 1 2
x 1
x 2
1
2
x x
2
2
4
4
2
2
0: .
2 x 1
x 2
x x 1 2
2
1
2
> 0
= 0
x x 2 x 1
x 2
2
2
2
2 x 1
x 2
x x 1 2
2
1
< 0
2
0
x x 2 x 1
x 2
2
2
a.c < 0
2
x 1
x 2
x 1
x 2
2 x 1
x 2
x x 1 2
0
2
2 x 1
x 2
0
0
P
0P
1; x2:
0
S
0
1 + x2
1 . x2
0
P
0
S
+
2
7/
4 + bx2 + c = 0
2 = t
0) 2 = bx + c (*) (a 0)
0
2 + bt + c = 0
< 0 at2 + bt + c = 0
2 = t
= 0 t 0.
> 0 t t =
