Ma Thematic A - 130 Problems in Geometry

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+ A 1 = 90 o

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= 90

.

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3 ( ) ) . , . // // ) // , , AB B . = + + = + + = 2 2 2 . 4 ( )

. ==90 . = = = = = = ---> //. ) // ( ) =/2 (1) , =/2 (2) ( ) =/2 (3) 1,2,3 =(++)/2

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100( Olympiad level) http://www.mathematica.gr/forum/viewtopic.php?f=50&t=4114

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= 20 (-3) (=) = 20 . , = 60 = 50 . . .

= 20O = ( = = 50 ) = = = 60O (==) = 40O = == = 70O

= 40 = 30O

2

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B1 = A1 B1 + B2 = 90 ,

1 + B2 = 90 , , E., 1-1 , :

1 = A3 1 + 2 = 90 , 3 + 2 = 90 , , . , , ( ) , .

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http://www.mathematica.gr/forum/viewtopic.php?f=27&t=261 (=90), G . , .

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1 , = . 3 3 2 2 2 2 = = , : 2 = = , 3 3 3 3 3 = (1) 3 . =

2 ++ + = = = = 2 3 3 3 2 2 = 2 2 = ( + )( ) = 2 ( ) = 2 ( ) (3) , : = (2)

(2)

5 4 + + 5 4 3 = 2 (2) (3) : = = , = 3 2 3 3: = = 2

3 3 4 3 5 3 2 = = = , = 3 2 2 3 2 3 2

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http://www.mathematica.gr/forum/viewtopic.php?f=20&t=1757 (.2-3), 1990, . 1. ( = ) , .A

.

20 30

140 10 40

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30 140 50 . : = 80

, : 30 80 80 80 80 = = = = =1, 140 50 2140 50 90 190 10

= , = = 70 .

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2 30 70 50 80 = = = 20 450 20 70 80 240 40 40 40 40 = = = = =1 450 20 70 220 70 50 90 50

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44 http://www.mathematica.gr/forum/viewtopic.php?f=50&t=62

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F Q H P

D

C M

E BFE = BAE BQP = BAE . BAQP . AD DCB AB + + AB + FDM 2 2 = AB + 360 = BQA = 2 2 4 BC DCA AB + + AB + ECN 2 2 = AB + 360 BPA = = 2 2 4 BAQP , BQP = BAE . BQP = BFE . QP EF . .

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: Tetragono III.png . : . : . : . = 1/2( + ) ^2 = 1/4( + )^2 = 1/4(^2 + 2*+^2) = 1/4(^2 + 2**1+^2) = 1/4(^2 - 2*+^2)= 1/4( - )^2 =1/4 ^2 . latex word xr.tsif R(,90 ) R(AB)= --- > = == , = (*) +=180 -- >+ =180 -- > =180 --> , . (#) =/2=/2 ( *) :

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( ) =// , =// , =

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, = =2.

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C , D : = 50 C = 100 .

D=10. /10=D/150=D/30 =2*D*10. '=10( ' ). AK'= (10*)/40. /' = 2*40*(D/AB) AD/AB=50/80 =40/80 /'=24040/80=1. =' ....

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y

(- , 80)

20 30

50

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P(0, 0)

30

B

x

: y = -80 x D x 140 C : y = -40x - 40 C C( x0 , y0 ) , : 1 2 40 x0 = y0 = 80 x0 1 + 2 40 ... y0 = 40 x0 40 y = 240 0 1 + 2 40

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1 2 40 240 : x + y = + = ... = 1 , (PC) = 1, DPC 2 2 1 + 40 1 + 40 2 0 2 0

2

2

40 DC. D 2 80 1 1 1 1 1+ + 80 = = 2 2 2 80 280 1 2 : ( x + 1) + y2 = 42 80 xx ( x0 , 0) .

(DA) = (PA) =

2

2

1 1 x0 + 1 = 2 4 80 280 1 1, 0 , D xx. 280 (D) = (DB) DA = DA = 50 AP= 30 .

: ( x0 + 1) + 02 =2

ABCD, . PBC= 30 . x 30. , , B :PB = 30 P = 70

. :1 1 PB P P 1 280 P = 280 = = 70 30 70 30 270 80 30 1 1 10 10 2 2 = = =1 60 80 1 10 2

() = (C) = 1, , C , C, PBC= 30 , ...

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71 http://www.mathematica.gr/forum/viewtopic.php?f=50&t=1649&start=0

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C , D : = 50 C = 100 . D=10. ( . D D) =10. CD ( KDB= KCB=70). D= KCD=40

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72 http://www.mathematica.gr/forum/viewtopic.php?f=50&t=1765 ( . ,

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MG 2 2

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1 5

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1 ,S = 4 10

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