AB − AC JJG JJJG

1

ﻪﺗﺎﻘﻴﺒﻄﺗ و ﻲﻤﻠﺴﻟا ءاﺪﺠﻟا

1ﻦﻳﺮﻤﺗ ﻦﻜﻴﻟ و ﺎﺜﻠﺜﻣ ABC

ﻒﺼﺘﻨﻣ I

[ ]

^BC

نأ ﺚﺒﺗأ

(

^{2 2}

)

1

AI CB ⋅ =

2

AB − AC JJG JJJG

ﻦﻳﺮﻤﺗ 2

ﺮﺒﺘﻌﻧ و ﺎﻴﻋﺎﺑر ABCD

ﻒﺼﺘﻨﻣ I

[ ]

^AC

و ﻒﺼﺘﻨﻣ J

[ ]

^BD

.

نأ ﻦﻴﺑ

2 2 2 2 2 2

4

2

AB + BC + CD + DA = AC + BD + IJ

ﻦﻳﺮﻤﺗ 3

ﻦﻜﻴﻟ و ﺎﺜﻠﺜﻣ ABC

ا ةﺮﺋاﺪﻟا ﺰآﺮﻣ O و ﻪﺑ ﺔﻄﻴﺤﻤﻟ

ﻒﺼﺘﻨﻣ I

[ ]

^BC

و ، ' ﺔﻠﺛﺎﻤﻣ A ﺔﺒﺴﻨﻟﺎﺑ A

ﺔﻄﻘﻨﻠﻟ .O

1 - نأ ﺚﺒﺗأ

' 0 ; ' 0

BA BA ⋅ = CA CA ⋅ = JJJG JJJG JJJG JJJG

2 - نأ ﺞﺘﻨﺘﺳا

2 2

' ; '

AA AC ⋅ = AC AA AB AB ⋅ =

JJJG JJJG JJJG JJJG

3 - ﺐﺴﺣأ '

AA AI⋅ JJJJG JJJG ﺔﻟﻻﺪﺑ

و AB AC

4 - نأ ﺚﺒﺗأ

2 2 2 1 2

' 4

AI + A I = OA −

2

BC

5 -

( )

^∆ دﺪﺣ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ﺚﻴﺣ ىﻮﺘﺴﻤﻟا ﻦﻣ M

2 2

OM − JJJJG JJJJG AM BM OA ⋅ =

ﻦﻳﺮﻤﺗ 4

ﻦﻜﻴﻟ و ﺎﺜﻠﺜﻣ ABC

( )

^∆

ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ﺚﻴﺣ ىﻮﺘﺴﻤﻟا ﻦﻣ M

( ) ( )

AC × AM ABJJJJG JJJG⋅ =AB× AM ACJJJJG JJJJG⋅ .

ﺮﺒﺘﻌﻧ ـﻟ ﻲﻠﺧاﺪﻟا ﻒﺼﻨﻤﻟا ﻊﻗﻮﻣ I

BACn

 

 

1 - نأ ﻦﻴﺑ

( ) ( )

^{AI ⊂ ∆}

2 - نأ ﺞﺘﻨﺘﺳا

( ) ( )

^{AI = ∆}

ﻦﻳﺮﻤﺗ 5

ﻦﻜﻴﻟ ﺎﺜﻠﺜﻣ ABC

1 - ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ دﺪﺣ ﺚﻴﺣ ىﻮﺘﺴﻤﻟا ﻦﻣ M

2 2 2 2

MA − MB = AB − AC

1 - ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ دﺪﺣ ﺚﻴﺣ ىﻮﺘﺴﻤﻟا ﻦﻣ M

2 2

AM − JJJJG JJJJG BM CM ⋅ = AB

ﻦﻳﺮﻤﺗ 6

ﻦﻜﻴﻟ عﻼﺿﻷا يوﺎﺴﺘﻣ ﺎﺜﻠﺜﻣ ABC

1 - نأ ﻦﻴﺑ

2

2 AB AC ⋅ = AB JJJG JJJG

ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ دﺪﺣ ﺚﻴﺣ ىﻮﺘﺴﻤﻟا ﻦﻣ M

MB MC MA ⋅ =

2

JJJG JJJJG

3 - ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ دﺪﺣ ﺚﻴﺣ ىﻮﺘﺴﻤﻟا ﻦﻣ M

2

2 MB MC ⋅ = AB JJJG JJJJG

4 - ﺮﺒﺘﻌﻧ

( )

^∆

ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ﺚﻴﺣ ىﻮﺘﺴﻤﻟا ﻦﻣ M

0 CA CMJJJG JJJJG JJJG JJJJG⋅ +AB AM⋅ =

أ - دﺪﺣ ﻊﻃﺎﻘﺗ ﺔﻄﻘﻧ D

( )

^∆

(

^BC

)

و

ب - نأ ﻦﻴﺑ

( )

M DM CB CA CM AB AM

∀ ∈ Ρ JJJJJG JJJG⋅ =JJJG JJJJG JJJG JJJJG⋅ + ⋅

د - ﺔﻌﻴﺒﻃ ﺞﺘﻨﺘﺳا

( )

^∆

ﻦﻳﺮﻤﺗ 7

ﺮﺒﺘﻌﻧ ﺎﺑﺪﺤﻣ ﺎﻴﻋﺎﺑر ABCD

(

^AB

) ( )

^{& CD} ﺚﻴﺣ و

ﻒﺼﺘﻨﻣ I

[ ]

^AC

و ﻒﺼﺘﻨﻣ J

[ ]

^BD

( )

^∆ و

ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ﺚﻴﺣ ىﻮﺘﺴﻤﻟا ﻦﻣ M

MA MC⋅ =MB MD⋅ JJJJG JJJJG JJJJG JJJJG

2 1 - نأ ﻦﻴﺑ

( )

^{2 2 2 2}

M ∈ ∆ ⇔ MI − MJ = IA − JB

2 - نأ ﻦﻴﺑ

( )

^∆

ﻰﻠﻋ يدﻮﻤﻋ ﻢﻴﻘﺘﺴﻣ

( )

^IJ

ﺔﻄﻘﻧ ﻲﻓ ﺎهﺪﻳﺪﺤﺗ ﺐﻴﺠﻳ E

.

ﻦﻳﺮﻤﺗ 8

ﻦﻜﺘﻟ و A

ﻦﻴﺘﻔﻠﺘﺨﻣ ﻦﻴﺘﻄﻘﻧ B

1 -

( )

^C دﺪﺣ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ﺚﻴﺣ M

2 MA = MB

2 -

( )

^E دﺪﺣ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ﺚﻴﺣ M

2 2 2

MA + MB = AB

3 - دﺪﺣ ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ F

ﺚﻴﺣ M

MA MB AB ⋅ =

2

JJJG JJJG

ﻦﻳﺮﻤﺗ 9

ﻦﻜﻴﻟ ﻲﻓ ﺔﻳواﺰﻟا ﻢﺋﺎﻗ ﺎﺜﻠﺜﻣ ABC

و A

( )

^C

ﻋﻮﻤﺠﻣ ﻂﻘﻨﻟا ﺔ

ﺚﻴﺣ M

2

6

2

AM + JJJG JJJJG AB MC ⋅ = AB

1 - دﺪﺣ و I ﻊﻃﺎﻘﺗ ﻲﺘﻄﻘﻧ J

(

^AB

) ( )

^C و

2 - نأ ﻦﻴﺑ

( )

^{2 6 2}

M MI MJ AM AB MA AB

∀ ∈ Ρ JJJG JJJG ⋅ = + JJJG JJJG ⋅ −

3 - ﺔﻌﻴﺒﻃ ﺞﺘﻨﺘﺳا

ﻳﺮﻤﺗ ﻦ 10

ﻦﻜﺘﻟ و A

و ﻦﻴﺘﻔﻠﺘﺨﻣ ﻦﻴﺘﻄﻘﻧ B

( )

^C

ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ﺚﻴﺣ M

2

2

2

3 0

AM + MB − MA MB × =

1 - نأ ﻦﻴﺑ

( )

^;

( )

B∉ C A∉ C

2 - نأ ﻦﻴﺑ MA

ﺔﻟدﺎﻌﻤﻠﻟ ﻞﺣ MB

2 3 2 0

x − x + =

3 - ﺔﻌﻴﺒﻃ ﺞﺘﻨﺘﺳا

( )

^C

.

ﻦﻳﺮﻤﺗ 11

ﻦﻜﻴﻟ و ﺎﺜﻠﺜﻣ ABC

و I و J ﻰﻟإ ﻲﻟاﻮﺘﻟا ﻰﻠﻋ ﻲﻤﺘﻨﺗ ﻂﻘﻧ K

[ ]

^BC

[ ]

^CA و

[ ]

^AB و

( ) ∆

1 و ﻢﻴﻘﺘﺴﻤﻟا

ﻰﻠﻋ يدﻮﻤﻌﻟا

(

^BC

)

ﻦﻣ رﺎﻤﻟا و وI

( ) ∆

2

ﻰﻠﻋ يدﻮﻤﻌﻟا ﻢﻴﻘﺘﺴﻤﻟا

(

^AC

)

ﻦﻣ رﺎﻤﻟا و و J

( ) ∆

3

ﻢﻴﻘﺘﺴﻤﻟا

ﻰﻠﻋ يدﻮﻤﻌﻟا

( )

^BA

ﻦﻣ رﺎﻤﻟا و K

1 - نأ ﻦﻴﺑ

( ) ^{∆ =}

¹

{ ^{M ∈} ( ) ^{P BM}

^{/ 2}

^{− CM}

²

^{= BI}

²

^{− CI}

²

}

2 - تﺎﻤﻴﻘﺘﺴﻤﻟا نأ ﺞﺘﻨﺘﺳا

( ) ^∆

₁

( ) ^∆

₂ و

( ) ^∆

₃ و نﺎآ اذإ ﻂﻘﻓو اذإ ﺔﻴﻗﻼﺘﻣ

2 2 2 2 2 2

0

BI − CI + CJ − AJ + AK − BK =

12ﻦﻳﺮﻤﺗ ﻦﻜﻴﻟ و ﺎﺜﻠﺜﻣ ABC

ﻪﻠﻘﺛ ﺰآﺮﻣ G

1 - نأ ﻦﻴﺑ

2 2 2

2 2 2

9

AB AC BC

GA = + −

2 - ﺐﺴﺣأ

( ^AC

²

^{− AB GA}

²

)

²

⁺ ( ^BA

²

^{− BC GB}

²

)

²

⁺ ( ^CB

²

^{− CA GC}

²

)

²

3 - ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ دﺪﺣ ﺚﻴﺣ M

( ^AC

²

^{− AB}

²

) ^MA

²

⁺ ( ^BA

²

^{− BC}

²

) ^MB

²

⁺ ( ^CB

²

^{− CA MC}

²

)

²

⁼

⁰

ﻦﻳﺮﻤﺗ 13

ﻦﻜﻴﻟ ﺚﻴﺣ ﺎﺜﻠﺜﻣ ABC

2 AB =AC = BC و

ﺢﺟﺮﻣ G

(

^{A; 1−}

) ( )

^B;1 و

( )

^C;1 و

1 - ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ دﺪﺣ ﺚﻴﺣ M

2 2 2

MB + MC = MA

2 - أ - نأ ﻦﻴﺑ

( )

²

M P MB MC MA AG

∀ ∈ JJJG JJJJG + − JJJG JJJG =

ﻂﻘﻨﻟا ﺔﻋﻮﻤﺠﻣ ﺞﺘﻨﺘﺳا ب ﻖﻘﺤﺗ ﻲﺘﻟا M

2 2 2 2 8 2

MB + MC − MA = BC