ﺔﻓﺪﮭﺘﺴﻤﻟا تاءﺎﻔﻜﻟا 12 12

تاءﺎﻔﻜﻟا

ﺔﻓﺪﮭﺘﺴﻤﻟا

ﻁﻘﻨ ﺔﻴﻤﺎﻘﺘﺴﺍ ﺕﺎﺒﺜﻹ ﻲﻜﺎﺤﺘﻟﺍ ﺹﺍﻭﺨ لﺎﻤﻌﺘﺴﺍ

.

لﺌﺎﺴﻤ لﺤ ﻲﻓ ﺔﻴﻁﻘﻨﻟﺍ ﺕﻼﻴﻭﺤﺘﻟﺍ ﻑﻴﻅﻭﺘ ﺔﻴﺴﺩﻨﻫ

.

ﻲﺴﺩﻨﻫ لﺤﻤ ﻥﻴﻴﻌﺘ

.

ﺔﻴﺴﺩﻨﻬﻟﺍ ﺕﺍﺀﺎﺸﻨﻹﺍ لﻭﺤ لﺌﺎﺴﻤ لﺤ

.

ﻲﻓ ﻢﻜﺤﺘﻟا ﻢﻠﻌﺘﻤﻟا ﻦﻣ ﺐﻠﻄﯾ ﻞﺼﻔﻟا اﺬھ ﻲﻓ :

Ã

ﺲﻜﻌﻟا و ﺔﯿﻋﺎﻌﺸﻟا ﺔﻗﻼﻌﻟا ﻰﻟا ﻲﻛﺎﺤﺘﻟا ﻒﯾﺮﻌﺗ ﺔﻤﺟﺮﺗ

ﺢﺟﺮﻤﻟا هﺰﻛﺮﻣ كﺎﺤﺘﺑ ىﺮﺧﻷا ةرﻮﺻ ﺎﻤھاﺪﺣإ نأ و ﻦﯿﺘﻄﻘﻧ ﺢﺟﺮﻣ ﻦﯿﺑ ﺔﻗﻼﻌﻟا ﺔﻈﺣﻼﻣ

Ã

ﮫﺘﺒﺴﻧ ﺪﯾﺪﺤﺗ ﺐﻠﻄﯾ و

Ã

ﻲﻛﺎﺤﺘﻟا ﻊﻣ تﻼﯾﻮﺤﺘﻟا هﺬھ مﺪﺨﺘﺴﺗ و ﺔﻋﻮﻨﺘﻣ ﻦﯾرﺎﻤﺗ لﻼﺧ ﻦﻣ ىﺮﺧﻷا تﻼﯾﻮﺤﺘﻟا ﻰﻄﻌﺗ

ﻧ ﺔﻋﻮﻤﺠﻣ ﻦﯿﯿﻌﺘﻟ ﺔﯿﺳﺪﻨھ تاءﺎﺸﻧإ ﻲﻓ مﺪﺨﺘﺴﺗ ﺎﻤﻛ ﺔﻨﯿﻌﻣ ﺔﯿﺻﺎﺧ ﻖﻘﺤﺗ يﻮﺘﺴﻤﻟا ﻦﻣ ﻂﻘ

تﺎﻤﯿﻘﺘﺴﻣ ةﺪﻌﻟ ﻊﻃﺎﻘﺘﻟا ، يزاﻮﺘﻟا ، ﺔﯿﻣﺎﻘﺘﺳﻹا تﺎﺒﺛﻹ ﻲﻛﺎﺤﺘﻟا لﺎﻤﻌﺘﺳا ﻲﻓ ﻢﻜﺤﺘﻟا ﻢﻠﻌﺘﻤﻟا ﻰﻠﻋ

Ã

ﺔﻄﻘﻧ ﻲﻓ

Ã

يﺮﻈﻨﻟا نﺎھﺮﺒﻟﺎﺑ ﺎھﺪﯿﻛﺄﺗ و تﺎﻨﯿﻤﺨﺗ ﻊﺿﻮﻟ بﻮﻠﻄﻣ ﺔﯿﻜﯿﻣﺎﻨﯾﺪﻟا ﺔﺳﺪﻨﮭﻟا تﺎﯿﺠﻣﺮﺑ لﺎﻤﻌﺘﺳا

12

لوﻷا طﺎﺸﻨﻟا :

فﺪﮭﻟا ﺔﻄﻘﻨﻟا ةرﻮﺻ و ﺰﻛﺮﻤﻟا ﺔﻓﺮﻌﻤﺑ ﻲﻛﺎﺤﺘﻟا ﺔﺒﺴﻧ ﻦﯿﯿﻌﺗ :

* OBuuur=k OAuuur

1 ( k = 1 2 3 ( k = 2 3 ( دﻮﺟﻮﻣ ﺮﯿﻏ k ﻂﻘﻨﻟا ﺔﯿﻣﺎﻘﺘﺳا مﺪﻋ )

(

* GDuuur=k GBuuur

1 2 ( k =3 2 ( 2 k = − 3

ﻞﻜﺸﻟا ﻲﻓ ﺢﯿﺤﺼﺗ ( )

ﺔﻄﻘﻨﻠﻟ يدﻮﻤﻌﻟا ﻂﺴﻤﻟا ﻲھ B ﻰﻠﻋ A

) GD ، ((

2 k =

ﻞﻜﺸﻟا ﻲﻓ * )

5 ﺢﯿﺤﺼﺗ : ( ﺔﻄﻘﻨﻟا :

ﻦﯿﻤﯿﻘﺘﺴﻤﻟا ﻊﻃﺎﻘﺗ ﻲھ M )

ON و (

1) d

، ( 2

k = −3

طﺎﺸﻨﻟا ﻲﻧﺎﺜﻟا : فﺪﮭﻟا ﻲﻛﺎﺤﺘﻟا ماﺪﺨﺘﺳﺎﺑ ﻂﻘﻧ ﺔﯿﻣﺎﻘﺘﺳا تﺎﺒﺛإ :

1

IF IE EF IF EF IB

(

IC IB BC IB BC IA

= = = + =

+

2

IB EF

(

IA = BC

3

2

(

IG =3ID uuur uur

نﻷ

2 3 IF FG IC =CD =

ﺚﻟﺎﺜﻟا طﺎﺸﻨﻟا :

فﺪﮭﻟا تﺎﺣﺎﺴﻤﻟا ﺮﺒﻜﯾ ﻲﻛﺎﺤﺘﻟا :

k2

ةﺮﻣ ) ﻲﻛﺎﺤﺘﻟا ﺔﺒﺴﻧ k نﺎﻛ اذإ (

1 k f

ﺎھﺮﻐﺼﯾ و

k2

نﺎﻛ اذإ ةﺮﻣ

1

k p

1

*

3 AM CN

AB =CB =

*

1 1

2 MM =3AA

و

1 1

1 PP =3AA

و

1

CN =2NB

1; 1; 1

ﻊﻣ

A M P

ﻂﻘﻨﻠﻟ ﺔﯾدﻮﻤﻌﻟا ﻂﻗﺎﺴﻤﻟا ﻲھ

; ;

A M P

ﻢﯿﻘﺘﺴﻤﻟا ﻰﻠﻋ ﺐﯿﺗﺮﺘﻟا ﻰﻠﻋ

) BC (

ﻊﺑاﺮﻟا طﺎﺸﻨﻟا :

فﺪﮭﻟا ﺮﻤﻟا ةرﻮﺻ ﺎھﺰﻛﺮﻣ ةﺮﺋاد ﻲھ ﻲﻛﺎﺤﺘﺑ ةﺮﺋاد ةرﻮﺻ :

ﺎھﺮﻄﻗ ﻒﺼﻧ و ﺰﻛ

k R

1

( AMBM′ ﻞﯿﻄﺘﺴﻣ

2 ) ( AM يزاﻮﯾ ( ) BN ( 3 ( 3 k = 4 ( O Nuuuur uuur uuur′ =O I′ +IN =k OMuuuur

ﺔﮭﺟﻮﻣ لﺎﻤﻋأ 1

: فﺪﮭﻟا ﻲﻛﺎﺤﺘﻟا لﺎﻤﻌﺘﺳﺎﺑ ﻲﺳﺪﻨھ ﻞﺤﻣ ﻦﯿﯿﻌﺗ :

1

1

(

IG =3IM uuur uuur

2

( ) ( )

( h C = C′

( )

^C′

ﻲﺗرﻮﺻ اﺪﻋ ةﺮﺋاد و A

ﺔﻄﺳاﻮﺑ B h

3

( ) ( )

( h O = O′ ﺚﯿﺣ

1 IO′ =3IO uuur uur

( )

^C′ ﺎھﺰﻛﺮﻣ O′

ﻞﻤﺸﺗ و G

لﺎﻤﻋأ ﺔﮭﺟﻮﻣ 2 : فﺪﮭﻟا ﻲﺳﺪﻨھ ءﺎﺸﻧإ ﻲﻓ ﻲﻛﺎﺤﺘﻟا لﺎﻤﻌﺘﺳا :

) 1 ﻞﯿﻠﺤﺘﻟا ﺔﻠﺣﺮﻣ ( * :

ةرﻮﺻ ﺔﻄﻘﻨﻟا ﻲھ K

C

* ةرﻮﺻ

ﻲھ I ﻊﻃﺎﻘﺗ ﺔﻄﻘﻧ E )

AI ـﻟ يزاﻮﻤﻟا ﻊﻣ ( )

LI ﻦﻣ مﻮﺳﺮﻤﻟا ( B

* ةرﻮﺻ

ﻲھ J ﻊﻃﺎﻘﺗ ﺔﻄﻘﻧ D )

AJ ـﻟ يزاﻮﻤﻟا ﻊﻣ ( )

LJ ﻦﻣ مﻮﺳﺮﻤﻟا ( C

*

ةرﻮﺻ J , I , L و

رﻮﺻ ﻲھ K D , E , B

و ﺐﯿﺗﺮﺘﻟا ﻰﻠﻋ C

) 2 ﺐﯿﻛﺮﺘﻟا ﺔﻠﺣﺮﻣ ( * :

ﺔﻟﺄﺴﻤﻠﻟ ﻞﺣ IJKL كﺎﺤﺘﺑ ﻊﺑﺮﻣ ةرﻮﺻ )

(

)

BE ، ( ) LI ، ( ) LJ و ( ) DC اﻮﺘﻣ ( ﺔﯾز AL AI AJ AK

AB =AE =AD = AC =k

, , ,

AL=k AB AI =k AE AJ =k AD AK =k AC uuur uuur uuur uuur uuur uuuur uuuur uuuur

ﻊﻣ J , I

, و K رﻮﺻ ﻲھ L D , E

, و C ﺐﯿﺗﺮﺘﻟا ﻰﻠﻋ B

* نﻷ ﺪﯿﺣو ﻞﺣ ﺪﯿﺣو BEDC

ﺊﻁﺎﺨ ﻡﺃ ﺢﻴﺤﺼﺃ :

ﻰﻟﺇ ﻥﻤ

لﺍﺅﺴﻟﺍ ﻡﻗﺭ

1 2

3 4

5 6

7 8

ﻡﻜﺤﻟﺍ ﺢﻴﺤﺼ ﺢﻴﺤﺼ

ﺊﻁﺎﺨ ﺢﻴﺤﺼ ﺊﻁﺎﺨ ﺊﻁﺎﺨ ﺢﻴﺤﺼ ﺢﻴﺤﺼ

ﺕﺍﺭﺎﻴﺘﺨﻻﺍ ﺓﺩﺩﻌﺘﻤ ﺔﻠﺌﺴﺃ :

ﻥﻤ ﻰﻟﺇ

لﺍﺅﺴﻟﺍ ﻡﻗﺭ

9

10 11

12 13

14

ﺍ ﺔﺤﻴﺤﺼﻟﺍ ﺔﺒﺎﺠﻹ

1 3

1 2

2 1

.(1 IA IB 2

=1 ، .(2 OP OQ=3 ،

.(3 AB IJ =−4 ،

.(4 CD AB=

.(1 ﺓﺭﻴﻅﻨ ﻲﻫ ﻰﻟﺇ ﺔﺒﺴﻨﻟﺎﺒ A

. B

.(2 ﺓﺭﻴﻅﻨ ﻲﻫ ﻰﻟﺇ ﺔﺒﺴﻨﻟﺎﺒ C

. D

.(1 k = -3 ، .(2 k = 5 ، .(3 k = 2 ، .(4 3 k = -2 .

ﺄﻁﺨﻟﺍ ﺏﻴﻭﺼﺘ )

ﺓﺭﻴﻅﻨ D' ﻰﻟﺇ ﺔﺒﺴﻨﻟﺎﺒ D

ﻥﺃ ﺕﺒﺜﺃ C ﻑﺼﺘﻨﻤ D'

] ([AF ﻥﻜﻤﻴ ﺱﻟﺎﻁ ﺔﻴﺭﻅﻨ لﺎﻤﻌﺘﺴﺍ .

ﻥﺃ ﺕﺎﺒﺜﺇ ﻥﻜﻤﻴ :

AECF ﻉﻼﻀﺃ ﻱﺯﺍﻭﺘﻤ

.

ﺄﻁﺨﻟﺍ ﺏﻴﻭﺼﺘ )

ﻊﻁﺎﻘﺘ ﺔﻁﻘﻨH )

(AB ﻭ ) (EI ﻥﺃ ﺕﺒﺜﺃ ﻑﺼﺘﻨﻤ H

] [EI ﺔﻘﻴﺭﻁ ﺱﻔﻨ ( .18

.(1 A'B'C'D'EFGH ، ﺏﻌﻜﻤ

.(2 ﺓﺭﻭﺼ ﻲﻫ H

ﺓﺭﻭﺼ ، O ﻑﺼﺘﻨﻤ ﻲﻫ F

] .[FB

.(1 ﺓﺭﻭﺼ ﻲﻫ C

، A .(2 BA BC 4

= 1 ، .(3 ، لﺎﺸ ﺔﻗﻼﻋ .(4

ﻡﻌﻨ .

ﺏﻴﻭﺼﺘ )

ﻉﻼﻀﻷﺍ ﺱﻴﺎﻘﺘﻤ ﺙﻠﺜﻤ ABC (

،

.(1 3 k1=2 ، .(2 ﺯﻜﺭﻤ h2

ﻊﻁﺎﻘﺘ ﺔﻁﻘﻨ ﻭﻫ )

(CB ﻊﻤ ) (NM ﻪﺘﺒﺴﻨ ﻭ 3

k2= -2

3 k = -2

.( 1 3 k = -1 ﻭﺃ k = -3

، .(2 3 k = 2 2 ﻭﺃ k = 3 ، .(3 2 k = 1 ﻭﺃ k = 2 ، .(4 k =

ﻲﻜﺎﺤﺘﻟﺍ ﺔﺒﺴﻨ 3

k = 1

.(1 4 ، 11 .(2

-2 ، .(3

3 ، 2 .(4

3

−1

k = -3

.(1 ، ﻻ .(2 ﻡﻌﻨ ) ﻱﺯﻜﺭﻤ ﺭﻅﺎﻨﺘ (

3 K = 2

.(1 ﻊﻁﺎﻘﺘ ﻭﻫ ﻲﻜﺎﺤﺘﻟﺍ ﺯﻜﺭﻤ )

(AB ﻊﻤ ) .(CD

15 9

14

1 8

18 19 20 21

24 25 26

29 30

31 32 16

17

22 23

.(1 ﺓﺭﻜﻓ ﺱﻔﻨ 32

، .(2 OA OB 3

= 4

.(1 ﺓﺭﻭﺼ A' ﻊﻁﺎﻘﺘ ﻲﻫ A

) (C ﻊﻤ ) (AB ، ﺓﺭﻭﺼ O' ﻁﻘﺴﻤ ﻲﻫ O

ﻰﻠﻋ A' ) . (PQ

) .(2 (D' لﻤﺸﻴ ﻱﺯﺍﻭﻴ ﻭ P )

(D ، ) (C' ﺎﻫﺯﻜﺭﻤ لﻤﺸﻴﻭ O'

. P

) .(3 (D' ﺱﻤﻴ ) (C' ، ) (C ﻭ ) (C' ﺎﻴﻠﺨﺍﺩ ﻥﺎﺘﺴﺎﻤﺘﻤ .

.(1 ﺽﺭﻔﻨ : AB AM =α ﺙﻴﺤ

] [

^0;1

∈ ﺩﺠﻨ ﻭ α : ' ' '

'M AB

A =α

.(2 ، ﻲﻠﺨﺍﺩﻟﺍ لﺩﺎﺒﺘﻟﺍ لﻤﻌﺘﺴﻨ .(3

ﺱﻟﺎﻁ ﺔﻴﺭﻅﻨ لﺎﻤﻌﺘﺴﺍ ﻥﻜﻤﻴ .

ﺓﺭﻭﺼ ﻲﻫ B

. C

ﻥﺎﺜﻠﺜﻤﻟﺍ ﻭ BDF

ﻥﺎﻬﺒﺎﺸﺘﻤ ACE .

ﺱﻟﺎﻁﻟ ﺔﻴﺴﻜﻌﻟﺍ ﺔﻴﺭﻅﻨﻟﺎﺒ ﺔﻨﺎﻌﺘﺴﻻﺍ ﻥﻜﻤﻴ .

،E

،B ﺭﻭﺼ F

،C

،O ﻟﺍ ﺍﺫﻬﺒ A' ﻭ ﻙﺎﺤﺘﺒ ﺏﻴﺘﺭﺘ ﻑﺼﺘﻨﻤ O

] . [AC

.(1 ﻥﻷ ) (DC لﻤﺸﻴ ﺓﺭﻭﺼ D ﻱﺯﺍﻭﻴ ﻭ B

) .(AB

.(2 ﻭ .(3 ﺱﻟﺎﻁ لﻤﻌﺘﺴﺍ .

ﺭﺒﺘﻌﻨ E1

، E2

، E3

ﺕﺎﻔﺼﺘﻨﻤ ]

[AB

، ] [BC ، ] [CD ، ﺏﻴﺘﺭﺘﻟﺍ ﻰﻠﻋ )

E1

، E2

، E3

ﺔﻴﻤﺎﻘﺘﺴﺍ ﻲﻓ (

G1

، G2

، G3

ﺭﻭﺼ ﻲﻫ E1

، E2

، E3

ﻩﺯﻜﺭﻤ ﻙﺎﺤﺘﺒ ﺘﺒﺴﻨ ﻭ O

3 ﻪ ﺔﻴﻤﺎﻘﺘﺴﺍ ﻲﻓ ﻲﻬﻓ 2 .

) .(1 (C' ﺎﻫﺯﻜﺭﻤ ﺓﺭﺌﺍﺩ



 



− − 2

; 3 ω 1 ﺎﻫﺭﻁﻗ ﻑﺼﻨ ﻭ r = 1

) ﺏﻴﺘﺍﺭﺘﻟﺍ ﺭﻭﺤﻤ ﺱﻤﺘ (

.(2

( )

¹

2 1 3

2

2  =



 

 + +

+ y

x

) .(1 (C ﺎﻫﺯﻜﺭﻤ ﺓﺭﺌﺍﺩ

( )

^0;0

ﺎﻫﺭﻁﻗ ﻑﺼﻨ ﻭ O r =

،2 ) (C' ﺎﻫﺯﻜﺭﻤ ﺓﺭﺌﺍﺩ

( )

^3;0

ﺎﻫﺭﻁﻗ ﻑﺼﻨ ﻭ A r' = 1

.(2 ﻥﺃ ﺎﻤﺒ : r' = 2+1= OA +

ﻥﺈﻓ r ) (C ﻭ ) (C' ﺎﻴﺠﺭﺎﺨ ﻥﺎﺴﺎﻤﺘﻤ .

.(3 2k=

−1

.(1 ﺕﻨﺎﻜ ﺍﺫﺇ ﻥﻤ ﺔﻁﻘﻨ M

) C1

( ﻥﺈﻓ ) (IM ﻊﻁﻘﻴ ) C2

( ﻲﻓ ﺙﻴﺤ N IN IM =− .

.(2 ﻠﺜﻤﻟﺍ ﺔﻨﺭﺎﻘﻤ ﻭﺃ ﻕﺒﺴ ﺎﻤﻤ ﺝﺎﺘﻨﺘﺴﺍ ﺕﺎﺜ

.

.(3 ﻥﺎﻔﺼﺎﻨﺘﻤ ﻥﺍﺭﻁﻘﻟﺍ .

.(1

°

=45 ˆC A ، B

°

=45 F B E) ) ﺎﻴﻠﺨﺍﺩ ﻥﺎﺘﻟﺩﺎﺒﺘﻤ .(

.(2 ، ﺱﻟﺎﻁ IC

IB 2

= 1 ، .(3 ﺓﺭﻭﺼ ﻲﻫ G ﻲﻜﺎﺤﺘﻟﺎﺒ D

، h ID IG 2

=1

.(1 ﻥﻴﺜﻠﺜﻤﻟﺍ ﻲﻓ ﻥﻴﻔﺼﺘﻨﻤﻟﺍ ﻡﻴﻘﺘﺴﻤ .

.(2 ﺓﺭﻭﺼ ] [BE ﻲﻫ ] [DG ، ) (PQ )⊥ (PN

.(1 ﺓﺭﻭﺼ ﻲﻫ A

، H .(2 ﺓﺭﻭﺼ ) (AI ﻭﻫ ) (IH ﺓﺭﻭﺼ ﻭ )

(AJ ﻭﻫ ) (JH

.(3 ﺭﻅﺎﻨﺘﻟﺍ ﺹﺍﻭﺨ

ﺓﺭﻭﺼ ﻲﻫ B ﻥﺍﺭﻭﺩﻟﺎﺒ C

ﻪﻨﻤ ﻭ r ﻊﻁﺎﻘﺘ C

(d1) ﻊﻤ (d'2)

ٍ ﺓﺭﻭﺼ (d2) ﺔﻘﻴﺭﻁﻟﺍ ﺱﻔﻨﺒ ﻡﻤﺘﻨ ﻭ .

ﻉﻼﻀﻷﺍ ﻱﺯﺍﻭﺘﻤ ﺹﺍﻭﺨ لﻤﻌﺘﺴﻨ .

ﺓﺭﺌﺍﺩ ) (c' ﺓﺭﻭﺼ ) (c ﻪﻋﺎﻌﺸ ﺏﺎﺤﺴﻨﺎﺒ .BA

ﻡﻴﻘﺘﺴﻤﻟﺍ )

∆' ( ﺓﺭﻭﺼ

∆) ( ﺔﻁﻘﻨﻟﺍ ﻰﻠﻋ ﺔﺒﺴﻨﻟﺎﺒ ﻱﺯﻜﺭﻤ ﺭﻅﺎﻨﺘﺒ ﻑﺼﺘﻨﻤ I

] . [AB

33 35

36 37 38 39 40 42 43

44

45

46

47

48 49

52 53 54

ﻡﻴﻘﺘﺴﻤﻟﺍ )

∆' ( ﺓﺭﻭﺼ

∆) ( ﺤﺘﺒ ﻩﺯﻜﺭﻤ ﻙﺎ ﻪﺘﺒﺴﻨ ﻭ A

.2

ﺓﺭﺌﺍﺩﻟﺍ )

(c' ﺓﺭﻭﺼ ) (c ﻩﺯﻜﺭﻤ ﻙﺎﺤﺘﺒ ﻑﺼﺘﻨﻤ O

] [AB ﻪﺘﺒﺴﻨ ﻭ 3

.1

.(1 ﺱﻟﺎﻁ ﺔﻴﺭﻅﻨ ﻕﻴﺒﻁﺘ ﻥﻜﻤﻴ .

.(2 ـﻟ ﻲﺴﺩﻨﻬﻟﺍ لﺤﻤﻟﺍ M₁

ﻭ M2

ﻥﻴﻌﻠﻀﻟﺍ ﺩﺎﺤﺘﺍ ﻭﻫ ]

[CD ﻭ ] [BE ﻥﻴﻌﻤﻟﺍ ﻥﻤ ﻩﺯﻜﺭﻤ ﻱﺫﻟﺍ BCDE

A

ﻥﺎﻜ ﺍﺫﺇ

≠0 ﻥﺈﻓ β GA GB β

−α =

.(1 ﻥﺎﻜ ﺍﺫﺇ

[ ]

^BC

A∉ ﻥﺈﻓ AB x= AC ﻥﺎﻜ ﺍﺫﺇ ﻭ ،

] [

^BC

A∈ ﻥﺈﻓ AB x=− AC

.(2 x = 2 ، x = -1 ، 2

x = 1

.(3 ﻥﻜﺘﻟ ﺓﺭﻴﻅﻨ D ﻰﻟﺇ ﺔﺒﺴﻨﻟﺎﺒ B

.A

ﺃ . ﻩﺩﺤ ﻱﺫﻟﺍ ﻡﻴﻘﺘﺴﻤﻟﺍ ﻑﺼﻨ ﻰﻟﺇ ﻲﻤﺘﻨﺘ C لﻤﺸﻴ ﻻﻭ D

.B

ﺏ .

ﺔﻌﻁﻘﻟﺍ ﻰﻟﺇ ﻲﻤﺘﻨﺘ C ]

. ]DB

ﺝ . ﺔﻌﻁﻘﻟﺍ ﻰﻟﺇ ﻲﻤﺘﻨﺘ C [

]DA ﻩﺩﺤ ﻱﺫﻟﺍ ﻡﻴﻘﺘﺴﻤﻟﺍ ﻑﺼﻨ ﻭﺃ لﻤﺸﻴ ﻻ ﻭ B

. D

.(1 ﺓﺭﻭﺼ ﻲﻫ M

، C .(2 ﻩﺯﻜﺭﻤ ﻥﺍﺭﻭﺩ r ﻪﺘﻴﻭﺍﺯﻭ A

2 π

) .(3 (B'C' ﺓﺭﻭﺼ ﻭﻫ )

(AM ـﺒ ﻪﻨﻤﻭ r ) (B'C' )⊥ . (AM

.(1

( )

^{A J}

h1 =

( )

^{A K} ، h2 = ،

.(2 BO k BI = ₁ ،

CO k CI = ₂

ﺝﺎﺘﻨﺘﺴﻻﺍ :

2 2

1+ = + = + = =

BO BC BO

CI BI CO

CI BO k BI k

.(3 OA OA k k OA k OA k IK

IJ + = ₁ + ₂ =( ₁+ ₂) =2

) .(1 (∆ ﻊﺒﺭﻤﻠﻟ ﺭﻅﺎﻨﺘ ﺭﻭﺤﻤ ﺓﺭﺌﺍﺩﻟﺍ ﻑﺼﻨ ﻭ ABCD

) (C لﻜﺸﻟﺍ ﺭﻅﺎﻨﺘ ﺭﻭﺤﻤ ﻲﻟﺎﺘﻟﺎﺒ ﻭ

∆)

⊥( ) (EF ﻭ

∆)

⊥( ) (DC ﻪﻨﻤﻭ ) )//(DC . (EF

.(3 5

= 5 OD ﻭ OE 5

= 5 OC ﻪﻨﻤﻭ OF h(C)=F .

KF=HE ﻭ

) )//(HF (KF ﻭ ) (HK )⊥ . (HE

لﻭﻷﺍ ﺀﺯﺠﻟﺍ .(1

AM AM 2

'=−1 ،

' 2 '' BM BM = .

.(2

BA AM

AM BA

BM ) 2

2 ( 1 2 2 '

' = + − =− +

.(3

Ω ﺔﻗﻼﻌﻟﺍ ﻕﻘﺤﺘ AB

A 2

−1

= Ω (*) ﺓﺩﻴﺤﻭ ﻲﻫ ﻭ .

(*) ﻰﻟﺇ ﻱﺩﺅﺘ

0 3ΩA−ΩB= لﺎﺸ ﺔﻗﻼﻋ

ﻥﻷ :

0 3 2 )

(3 2 3 2 3

3

3ΩA−ΩB= ΩB+ AB−ΩB= ΩB+ BA= AB + BA=

.(5

ﺎﺒ ﻥﻴﻟﺍﺅﺴﻟﺍ لﺎﻤﻌﺘﺴ (2

ﻭ (4 ﺩﺠﻨ M B

A M

M =−Ω + Ω −Ω =−Ω

Ω " 3

.

ﻱﺃ :

M"

ﺓﺭﻭﺼ ﻲﻫ ﻩﺯﻜﺭﻤ ﻙﺎﺤﺘﺒ M

ﻪﺘﺒﺴﻨ ﻭ Ω ) -1

ﻱﺯﻜﺭﻤ ﺭﻅﺎﻨﺘ .(

ﻟﺍ ﺀﺯﺠﻟﺍ ﻲﻨﺎﺜ .(1 h1

ﻩﺯﻜﺭﻤ ﻙﺎﺤﺘ ﻪﺘﺒﺴﻨ ﻭ G

2

−1 ، .(2 h2

ﻩﺯﻜﺭﻤ ﻙﺎﺤﺘ ﻪﺘﺒﺴﻨ ﻭ M

. 2 ، .(3 ﺎﻨﺘ ﻱﺯﻜﺭﻤ ﺭﻅ

.(4

ﻥﺃ ﺞﺘﻨﺘﺴﻨ ] :

[AP ، ] [BQ ، ] [CR ﺭﻅﺎﻨﺘﻟﺍ ﺯﻜﺭﻤ ﻲﻫ ﺓﺩﺤﺍﻭ ﺔﻁﻘﻨ ﻲﻓ ﻊﻁﺎﻘﺘﺘ .

.(1 AI AJ =2 ،

AJ BK =3

55 56 57

58 59

60

61

62

63

64

ﺎﻫﺭﻁﻗ .6

لﻭﻷﺍ ﺀﺯﺠﻟﺍ :

ﺭﻴﻟﻭﺃ ﻡﻴﻘﺘﺴﻤ

.(1 ﺭﻭﺼ

،A

،B ﻲﻜﺎﺤﺘﻟﺎﺒ C ﻲﻫ h

،A'

،B' . C'

.(2 ﺙﻠﺜﻤﻟﺍ ﺓﺩﻤﻋﺃ ﺭﻭﺼ ﻲﻜﺎﺤﺘﻟﺎﺒ ABC

ﻩﺭﻭﺎﺤﻤ ﻲﻫ h .

.(3 ﺭﻭﺼ ﻲﻜﺎﺤﺘﻟﺎﺒ H ﻲﻫ h

.O

.(4 ، O ، G ﺔﻴﻤﺎﻘﺘﺴﺍ ﻲﻓ H .

ﺍ ﺀﺯﺠﻟﺍ ﻲﻨﺎﺜﻟ : ﺭﻴﻟﻭﺃ ﺓﺭﺌﺍﺩ

) .(1 ﻁﻴﺤﻤﻟﺍ ﺓﺭﺌﺍﺩﻟﺍ ﻲﻫ (C' ﺙﻠﺜﻤﻟﺎﺒ ﺔ

A'B'C' ﺎﻫﺯﻜﺭﻤ ω)

.(

.(2 GO

G 2

−1

= ω ﻱﺃ OG O 2

= 3 ω ﻱﺃ OH O 2

=1 ω ω ﻱﺃ ﻑﺼﺘﻨﻤ ]

[OH

.(3 ﺭﻭﺼ ) (C ـﺒ ' ﺎﻫﺯﻜﺭﻤ ﺓﺭﺌﺍﺩ ﻲﻫ h ) ω

OH H 2

=1 ( ω ﻭﻫ ﺎﻫﺭﻁﻗ ﻑﺼﻨ ﻭ 2

ﺓﺭﻭﺼ ﺎﻬﺴﻔﻨ ﻲﻫﻭ r )

(C ـﺒ . h

.(4 ﺝﺎﺘﻨﺘﺴﻻﺍ ، ﺱﻟﺎﻁ ﻕﻴﺒﻁﺘ

A : H A ω ω '= ﻪﻨﻤﻭ ) (C'

A∈ ﺔﻘﻴﺭﻁﻟﺍ ﺱﻔﻨﺒ H HB

C ، ﻰﻟﺇ ﻥﺎﻴﻤﺘﻨﺘ H )

.(C'

.(5 ﺙﻠﺜﻤﻟﺍ ﺱﻭﺅﺭ ﺭﻭﺼ ﻲﻜﺎﺤﺘﻟﺎﺒ ABC

ﻲﻫ h' : H1

، H2 3 ، ﺕﺎﻔﺼﺘﻨﻤ H ]

[AH ، ] [BH ، ] [CH .

.(6 ﻊﺴﺘﻟﺍ ﻁﻘﻨﻟﺍ لﻤﺸﺘ ﺎﻬﻨﻷ

،A'

،B'

،C' HA

،

65