# Γεωμετρια Α λυκειου σημειώσεις

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• 1. - 1 :: http://mathhmagic.blogspot.com :

2. - 2 :: http://mathhmagic.blogspot.com , , . . , , - - . , . ' , . '. . . , . . . . , . . 1 . . 2. . A3. , . 4. . 3. - 3 :: http://mathhmagic.blogspot.com . . , , ( ) , , : , : = , : < , > . , AM = MB. . , . 4. - 4 :: http://mathhmagic.blogspot.com , . . KM : = (1) (1) = 1 KM (2) > , , : () = KM = ( , KM < ). : =-=- . . . 5. - 5 :: http://mathhmagic.blogspot.com , , : ()= () To (AB) . , , , () = . . ( 0). , , = 8 7 , , , 8 1 1 7 7 = + : = 1. + 1 7 1 7 . 8 7 = 1,14 . . - i) , , : = ,= = - = ii) =+=+= 1) ,,, , . : i)= ii)KM= 6. - 6 :: http://mathhmagic.blogspot.com = + + = 2 2 + + = 2 2 2 2 2 2 + + + + + = = + + + + = = = 10 3 2 = : 3 (10 ) .... 6 2 = = . =6 =4 2) ,, . , 1 ( ) 2 = + 3) 10 3 2 = , . 1. , ( ) . 2. , =. 3. , : ) . . ) . ) . 4. . . . 7. - 7 :: http://mathhmagic.blogspot.com 5 7 = , . =- =-, : 5 5 ( ) ( ) 7 7 7 7 5 5 7 5 5 7 5 7 12 5 7 12 12 = = = + = + = + = + i) To O 2 2 2 2 2 2 2 2 2 = + = + = + + = + + + = + = ii) To O A 2 2 2 2 2 2 2 2 2 = + = + = + + = + + + = + = 4) 5 7 = . , 5 7 12 12 = + 4) . , , 2 + = 8. - 8 :: http://mathhmagic.blogspot.com = , =y ==+y : 2 2 2 2 2 2 2 2 2 2 2 ( ) (2 ) 4 4 4 4 2 (1) y y y y x xy y y x xy y + = + + + = + + = + + + = + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2( ) 2 2( 2 ) 2 2 4 2 4 4 2 (2) x x y x x xy y x x xy y x xy y + = + + = + + + = + + + = + + (1) ,(2) . (1) = = += (2) + = += (3) + = (2) (3) : (1) + + + = + + = + ( ) + = = + 5) ,, : 2 2 2 2 2 2 + = + 6) : = , * . : = + 9. - 9 :: http://mathhmagic.blogspot.com = ()=() ( ) ( ) = 1 1 0 < < > ( ) (1 0 = = = = + = + : ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 ( ) P ON OM OA AP OB BM OA AN AB B OA OB OA AB B OA OB OA OA OB OA OA OB OA OB + + = + + + + + = + + + + + = + + + + + = + + + + = + + + = + + 7) ,, , = = >0. : (1 ) = + 8) , , P , . : P ON OM OA OB O + + = + + 10. - 10 :: http://mathhmagic.blogspot.com 1) , , : = 2 5 . 2) ) ) ) 3) . 5; 4) , = ++ 10 3 10 1 3 . 5) , = = . - . - .. - . () . . - . - . , , . . . : . . 27. - 27 :: http://mathhmagic.blogspot.com = . - : . . . . - , . , < , B + . 6)i A . . N < . 7) . N < . 8) = 2 , > + . 2 < 2 = 10) = = , = . 11). = > , > . 12.) < AM, < 90 >90. 13). , . N . . (. ) = , = = . . : . . . (, ) = , : < < . (, ) . . . . : . , , = : (, ), : 66. - 66 :: http://mathhmagic.blogspot.com ) , ) . , = = , ., OB , , , = . = = . 1) (, R) . , . : ) = + ) - = - 2) , . ) , . ) = + . 3) = AM=, . 4) (, ) . = , = 67. - 67 :: http://mathhmagic.blogspot.com 5) . . . : ) , ) = , ) + . 6) = MB. : ) AMO O MB = ) = 68. - 68 :: http://mathhmagic.blogspot.com d. = d . (, R1) (2, R2) R1 > R2 : . d>R1+R2 dR1-R2 ( ) d>R1+R2( ) 69. - 69 :: http://mathhmagic.blogspot.com , . . . . . 1 = 1 2=2, 12, , . . . . , : ) . . ( ), . ) . , . ) . . 70. - 70 :: http://mathhmagic.blogspot.com ) . , . ) . . 71. - 71 :: http://mathhmagic.blogspot.com 1) . , , , . 2) ( = . 3) (1, 1) (2,2) , 12 . (1, 1), (2, 2) 4) (K1) (K2) 3 1 . 5) . . =. = . 6) (, R) (. ) . . , : ) , . ; , ) < 2. 7) (, ) (, R) < R . KB (, R), (, ) , ) = . 72. - 72 :: http://mathhmagic.blogspot.com ) (, R) . (, R) KB , : i) , ii) = , iii) = . ) < R > R. 8) (, ) (, ) =. , , . 9) (, ) (, R). . (, ). (, R). 10) (, ) (, ), . 73. - 73 :: http://mathhmagic.blogspot.com : ) , ) , ) , . 1 2 , : ) , , 1 2 . ) , , . ) . ) . : ) . ) . ) , .. ; : . , //. , 74. - 74 :: http://mathhmagic.blogspot.com : 1 2 . = ( = ), 1 // 2 () = ( = ), 1 // 2 ( 1) 0 180 + = , 1 // 2 ( 2 ) 1 2 , 1 // 2 ( 2 ) : . : 1 // 2 1 2 , : ) , = = ) , , = = ) , 0 180 + = . 1 ) , , . ) , . 2 : . 1 2 1 2 75. - 75 :: http://mathhmagic.blogspot.com 1 // // 2 , 1 // 2 . 1 // 2 1 , 2 . 1 , 2 . . , 0 1 1 180 + < , 1 2 . 1 2 180, . 76. - 76 :: http://mathhmagic.blogspot.com , : ) , ) , ) () , ) , ) , ) ) , ) , ) ) 1 : , . 2 : , . ( ) ' ' y y x x ' ' y y x x '// ' Bx Ax By Ay '// ' Bx Ax By Ay 77. - 77 :: http://mathhmagic.blogspot.com (). . . . . . . . . 78. - 78 :: http://mathhmagic.blogspot.com : 0 180A B+ + = A B+ + = 2 : . : = = + 0 180 > + (1) . 3 : 4 : 60 - - : - 2 -4 =6 , , 2.6-4= 8 720 : . 1: . 2 79. - 79 :: http://mathhmagic.blogspot.com - : - 4 360 0 90 2 + = 0 90 2 + 2 . 2 = 0 90 2 . 0 90 2 = 0 180 + + = (1) 0 90 2 2 2 + + = (2) 0 0 0 90 (3), 90 90 (4) 2 2 2 2 2 2 2 + + = = = (1) ,(2) (3) (4) . (2) (4) . . : = + , = + , = + , : 0 0 360 , 180 (5) 2 2 2 2 + + + = + + + = (5) : 0 0 180 , 180 2 2 2 2 2 + + + + = = , : 0 0 90 180+ + = 0 0 0 90 , 90 , 90 + = = = . 80. - 80 :: http://mathhmagic.blogspot.com - //. // , // = = ( ) (1) = = (2) (1) (2) : + = + + = . , ( , 180 = + = ) . 0 0 0 0 0 0 180 5 20 2 20 180 7 180 25 + = + + = = = 0 0 0 5 25 20 145 = + = 0 0 2 25 20 35 = = . 0 145 = = = = , 35 = = = = . 1) //. = + 2) 1 2 1 // 2 0 5 20x = + , 0 2 20x = 81. - 81 :: http://mathhmagic.blogspot.com // , . : 0 60 = 0 60x = 0 0 0 110 60 50y = = . y A 0 0 0 130 50 180A y+ = + = //. // // //. , 1 1 = . 1 2 = . // , 1 1 = ( ) 1 2 = ( ) (1) 1 1 = . 3) 0 130 = , 0 120 = 0 120 = // 4) OB . // 82. - 82 :: http://mathhmagic.blogspot.com = = = . , : = (1) , = = . , = (2) (1) (2) = . 5) , . , . =. =180 140 = 40 , =60 =180 40 60 =80 . , : 1 = 2 B =30 1 = 2 =40 . =180 1- 1=1

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