Β ΓΕΩΜΕΤΡΙΑ ΣΗΜΕΙΩΣΕΙΣ

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  • - :

    1

    : AB = =

    .

    . . ==

    ,,,. . .

    ( ) , . :

    =

    =

    . , :

    =

    .

    .

    == ,

    == ,

    ==

    =

    = ,

    =

    =

    ++++++

    ====...

    ......

    A

  • - :

    2

    ,

    . , :

    =

    .

    .

    . . : . = 1.

    .

    1

    : .

    2

    : .

    3

    : .

  • - :

    3

    9o

    ( , ), ( , , ).

    . , . , , , .

    . . , 0.

    . ,

  • - :

    4

    2 = 2 = 2 = 2 = 2 = 2 + 2 2 = 2 + 2 2 =

    2 = = =

    ,

    , 90 = . , , 90 == = 1 . : 1

    =

    , =2 //

    o

    .

    , 90 = : =2 =2

    //)( 2

    22

    =+=

    =+=+

  • - :

    5

    .

    , =, =. : 2=2+2=2+2=2. =, ,

    , ,

    == 90 .//

    .

    , 90 = .

    1 = .

    =

    =2 // 1

  • - :

    6

    ()

    ()

    .

    90 ++= 2222

    90

    //

    :

    , ,

    .

    90222 =+= o

    90222 +> o

  • - :

    7

    .

    . 2

    22

    222 +=+ .

    . 1 2 . = 222 //

    1

    .

    += 2222 (1) ++= 2222 (2) =

    22)

    2(2222

    22222222 +=

    +=+=+

    2 .

    (1) (2) =2

    =

    == 422 2 //

    .

    : 4

    22 2222 +

    =

    :

    2 2 22 2 2

    4

    +

    = , 2 2 2

    2 2 2

    4

    +

    =

  • - :

    8

    ( , R) =

    , .

    : 22 = , . 22 = ,

    .

    : = = !

    =

    = :

    = + = +

    = - =

    = ( + )( )= 22 .

    : .

    1 2 :

  • - :

    9

    21 = (),

    = , ( )

    = ( ). B

    =

    = .

    : = + = + = - = = ( + )( ) = 22 .//

    ( , R ) = 22 ( , R )

    22 >0

    22 =0 22

  • - :

    10

    . . .

    1

    ( 90B

    < ) : 2-2=2-2

    ,

    :

    2-2=(2+2)-(2+2)=2+2-2-2=2-2.

    2

    84 37.

    ++=84 , =37. +=47. (1)

    2+2=2=1369 (2)

    (2) (1) : (47-)2+2=1369 ,

    2-47+420=0 12 35.

    , =35 =12 =12 =35.

    3

    90A

    = + 2 . .

    + 2 (+)2( 2 )2 2+2+222 2+2+22(2+2)

    2+2+222+22 2+2-2 0 (-)20 , .

  • - :

    11

    4

    90A

    = 2+

    2+2=

    3

    22.

    =2

    a , 90A

    = .

    2

    4

    a+

    2+2=

    3

    22

    2+2=

    25.

    4

    a

    2=2+

    2

    2

    2=2+

    2

    2

    .

    2+

    2= 2+ 2+2 2

    4

    +=2+

    2

    4

    a=

    25.

    4

    a

    5

    , 2+2=2+2.

    .

    :

    2=2+2 2=2+2.

    : 2+ 2= 2+2 +2+2= (2 +2)+( 2+ 2)=2+2.

    6

    =4 , =10 5.

    == . (),

    =. , 2+=

    2+= ( =) , = 3.2

    =

  • - :

    12

    2=2-2=16 => =4.

    2=2+2 =>== 49 16 65.+ =

    7

    2=2+2- (1) , 60

    = .

    1

    . : 2=2+2-2 (2). (1) (2)

    1 ,

    2+2-= 2+2-2 =>=1

    2 => 60

    = .

    2

    . 2=2+2-.

    =2

    2=2+2-

    2= 2=2+2-, .

    8

    ( 90A

    = ) = , = 3.

    ,

    .

  • - :

    13

    =2=2.

    2=(2)2-( 3 )2=2 , == ,2

    30o

    = 60 .oB

    =

    = 3 3

    .2 2

    = = =

    9

    =3 5

    2

    = ,

    ) =2 ) 90o

    > ) 2 , .2

    = =

    )

    ( )22 2 22 2 2 2 2 2 22 2. 3 22 2 5 25 18 24 2 4 4 4

    + + + = = =

    2 22 8 2 . = =

    ) >> 2=92 , 2+2=(2)2+2 2+2=52 , 2>2+2 90o

    > .

    ) 2 2 2

    2 2 2

    4

    +

    = =2 2 2

    2 218 2 4 4 24

    + = =

    2 2 2 2 2 2 2

    2 2 2 8 2 9

    4 4 4

    + +

    = = = .2

    =

    10

    A

    =120. 2=

    2 2.

    4

    2=2+2-2120 =>

    2=2+2-2(1

    2 ) =>2=2+2+ => 2+2=2- (1).

    1 2 2 2

    2 2 2

    4

    +

    = :

    2 2 2 2 2 2 2 2 2 2(1)2 2 2 2( ) 2( ) 2 2

    4 4 4 4

    a a

    + + = = = = =

    2 2.

    4

  • - :

    14

    11

    (,) . , 42=2+32.

    90

    = , .

    2+2=2 =>42+2=2 (1)

    2+2=2 , (1)

    2 22 2

    4

    + = => 2-2+42=42 =>

    42=2+32, .

    12

    . 2+2+2+2=22. 1 :

    2+2=22+2

    2

    (1).

    1 :

    2+2=22+2

    2

    (2).

    (1) (2) : 2+2+2+2=2(2+ 2)+ 2

    2

    +

    2

    2

    =>

    2+2+2+2=2(2+ 2)+ 2 (3).

    2=(+)2=2+2+2.

    ===== 2+2+2.=

    2+2+22

    .

    2

    = 2+2+

    2

    2

    => 2+2= 2-

    2

    2

    .

    (3) 2+2+2+2=2(2-2

    2

    )+ 2 =22-2+

    2=22.

  • - :

    15

    13

    ,, 4 ===.

    2+2=2+2+23

    .8

    ==== ,4

    a =.

    1 : 2+2=22+2

    .2

    a(1).

    . 1 :

    2+2=22+

    2

    22

    a 2+2=22+

    2

    8

    a 2=

    22 2

    8 ,2

    +

    (1) 2+2= 2+2-2 2

    8 2

    a a+ = 2+2+

    23.

    8

    a

    14

    =3,=8

    =60. .

    :

    2=2+2-2 => 2=64+9-2.24.1

    2 => 2=49

    => =7. =7.

    ( )

    2=2 2 2

    2 2 2 2.9 2.49 64 13 13.4 4

    AM

    + +

    = = = = =

    15

    =4 , =3 (,) .

    . =7

    8,

    )

  • - :

    16

    ) .

    ) 2 2-2=2. =>

    2 2

    4.2

    = =

    ) .=. =. 4

    =

    (1).

    2 2 2

    2 2 2

    4

    +

    =

    2 2 2 2 2 22 2 2 2.4 2.3 4 34

    4 4 2

    + + = = = (2).

    (1) (2) =4.2 8 34

    .3434

    =

    16

    ,,

    22 .

    4

    >

    2

    2

    4

    > 2 2 2 22 2

    4 4

    + > 2>22+22-2-4

    22>22+22-4 2>2+2-2 2>(-)2 >|-| ,

    ( ).

    17

    . 2+2=2+2

    . 1

    2+2=22+2

    2

    2+2=22+2

    2

    B. =

    , , 2+2=2+2.

  • - :

    17

    18

    = + . (1)

    + =

    22

    222222 ++

    +a =

    22

    222222 ++

    +a=

    22.

    2

    aa

    a=

    19

    . .

    . . ,

    90

    = . , 2=2+2 (1)

    , 2=2 2 2

    .2 2 4

    = =

    , :

    2 2 22 2 2 2 21 1 1 3( ) [ ] .

    2 4 4 4 2 16

    a aMN a

    = = = = =

    (1)

    2=2 2 23 7

    ,4 16 16

    a a a+ = =

    7.

    4

    a

    20

    . . 1 :

    2+2=22+2

    2

    ,

    2+2=22+2

    2

    B.

  • - :

    18

    : 2+2+

    2+2=22+2

    2

    +22+

    2

    2

    B=2(2+ 2)+

    2 2

    2

    +. = ,

    =2

    =

    2

    B.

    2+2+2+2=2+2.

    21

    : >

  • - :

    19

    2+ 2=2 2 +2+ 2-2.-2.= 2+ 2+2 (--)= 2+

    2+2.= 2+ 2+2.

    :

    90A

    > , 90 .B

    >

    24

    =8,=6 =5. ) ) .

    )