γεωμετρια α λυκειου βοηθημα 157-174
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Transcript of γεωμετρια α λυκειου βοηθημα 157-174
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8/11/2019 157-174
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158.
.
.
. ,
: ( )1K
A 12
= ( )K
22
= .
1 2
360 180
2 2
+
+ = + = + =
.
( )A 180 1+ = .
, ,
,
: ( )A 180 2+ = . (1), (2)
= , .
.
.
,
= . 180 (1)+ = ( )
180 (2)+ = ( . ).
(1), (2) + = + = .
,
.
1
2
B
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159.
(1)
=
. ( ) 180 2+ = .
( ) ( )1 , 2 180+ = , .
.
. = .
,
.
(1)= .
. ,
, ,
.
(2)= .
(1), (2) = , .
.
. 180 (1)+ =
,
. (2)=
. (1), (2) 180+ = ,
.
, .
.
180 (1)+ = . : + = + = ,
.
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160.
1. , . , . .
(1)=
. 2+ = . (2)
. (1), (2)
2 .+ = .
2. (,R).
, , , .
x= = .
o E x x 180+ + = ( )o E 2x 180 1 + = . K 2x=
( ) (1) :
o E K 180+ = , .
3.
, , . .
: 180+ = .
90=
90+ = .
( ) 90= = = , 90+ =
, 90+ = .
4.
.
, , ,
y = .
( ) 90= = ( ) x 1= . ( )y 2= .
x = ( ) (1), (2) y = .
A
B
x y
xx
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161.
i)
.
ii)
.
-
()
1. .
2. .
3. ,
.
. :
()
1.
.
2. .
+ = +OA , OB ,O :,
O
1
112
2
2
2 1
1 1 1 1 2 2 2 2
0 0
, , ,
180 , 180
, , ,
= = = =
+ = + =
= = = =
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162.
.
1
, , ,
, .
:
i. = =
ii. = = , =
iii. = -
i. = = = = ( ) = + =
= + .
:
2 2
+ = + = = =
: , = = = = .
ii. , = + = + = .
2 2 2 = + + + = + + + = + +
2
+ += = ( ) = = =
iii. ( ) = = = = + =
2
A . ,
.
. -
.
1 1 = ,
- 2 1
= , -
.
1 2
= ,
. .
O1
O
KB H
Z
A
E
M
K
B
Z
A
P
O
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163.
3
. ( Simson).
, E A ,
.
1 1 = 2 2
= . 1 2
= .
1 2
180 180
+ = + =
, .
4
(,R) .
.
, , : ) // , ) ' = '' .
) ( )
1 90= =
:
1 = 1 1
= ( ).
: 1 =
1 1 1
, // = = .
) : 1 1 1 = = = = .
5 (,R) (,) . ,
, : -
.
, . -
: 2 2 // + = + = .
K
A
A'
B
B'
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164.
. =
. -
. = = ( -
).
,
//2
+= (1).
2=
2=
2
+= (2).
+ = + , .
6
( = 90 ) , : + = 2 + .
, :
( ) 90 = = = = = . = = . = = (
).
( ) 2 2 + = + = + + + = + + = + .
7
(,).
,
. .
= ,
.
= , .
= , -
,
.
x
A
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165.
8
( ) = .
. ,
.
(,).
,
.
= = , .
= = = ( :
). = ,.
.
9
.
,
,
.
= .
= , //.
, = . = .
.
10
, ,
,
.
, :1 2
2= = .
1 2
2= = ,
E
'
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166.
. :
= AB .
2
=
.
:
2 2 2
180 90
2 2 2 2 2
+ = + = + = + = = =
180 180 90 180 90 902 2 2
= = = + =
902
= = ,
.
11
. -
, .
.
= = , -
.
// =
12 ,
. = .
.
( ) 1=
:
1 1 90 90 + = = (1)
( ) 1
= :
A
B
M
1
1
2
2
A
B
H
E
1
1
1
Z
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167.
1 90 = (2)
H
, ,
(1) (2)
1 1 1 1 1 90 = = = .
.
13
,
.
.
1 1 = ,
1 = , . 1 = ,
,
.
14 .
.
.
, 90 90 180+ = + = .
180+ = (1)
( ) 90 90 180+ = + = , .
= (2).
, :
90= + = + (3) :
( ) ( )(2) (3) (1)
90 180+ = + + = + + = + =
, .
B
M
1
1
A
B
Z
E
M
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168.
15
xO .
,, , x, O, :
. , , , ,
. = .
. , -
, , , -
.
, , , , .
. 1 2
= , AB A= ,. = .
16
(, R) (, ) R > .
(,) (, )
(, ) , .
.
: =
, -
: = + (1)
-
, . = -
(, ).
= (2) - .
= (3) -
.
(2 )
(3)
(1) = + =
17
B
, M. ,, ,
, , .
B
A
P
xy
O
K
12
E
B
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169.
, . :
1 1
= = = = , 2 2 = = = . //
// ,. .
18
, :
. , .
. .
. ,
. :1 2
2= =
1 2
2= = .
:
, :
1 1
2 2 2
+= + = + =
2 3 2 2
2 2 2
+= + = + = + = ,
3 2 = . =
, : = (1)
. 1 2 , , , = (2)
(1) (2) : = = ,
, .
19
. = .
A
I
1
1
2
23B
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170.
. :
= = , 1 2
= ( 1 =
, 2
= ).
= .
20
.
, , -
.
A B
.
2= = . -
, 1 = (1)
.
1 :
(1)
1 1 1 2= + = (2).
90 90 180+ = + = ,
, . 1 = (3),
.
.
2
= = . . 1 1 =
(4) . 1
:
(1) (3)
1 1 1 1 1 1 1 2 = + = = . ,
.
A
B
E
M
N
H
1
1
1 1
1
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171.
.
1. . ,
,
, :
.
.
2
. , , ( Euler)
. = 2.
2. , ,
.
NM N N= = .
3. -
.
4. -
. ,
.
. ,
, -
.
5. -
.
,
. .
M
K
K
B
H
Z
A
P
P M
O
K
M
NT
K
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172.
6.
.
.
7.
. -
.
8. (,).
, .
.
9. ,.
,
,
.
10. , -
, .
.
'
'
-
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173.
11. (,).
, .
, -
.
12. (,)
.
, ,
.
13.
,,,, -
. .
14. -
, , , , , :
. .
. , .
.
( = 90 )
, , .
, :
)
) , , , , .
O
M
K
A B
O
A
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