الجدع مشترك علمي - المغرب -

1. http://ad2math.voila.net/ adnanemath@gmail.com ‫ط‬ ‫أآ‬ ‫ن‬ .‫ذ‬ Page 1
‫ب‬ ‫ا‬‫أ‬
‫ا‬1:ABC،
1(‫أ‬I:AI=3AB
2(‫أ‬J:AJ 2AC= −
3(‫أ‬F:BF BA BC= +
4(‫أ‬K:AK AI AJ= +
5(‫أ‬L:
3 2
AL AB AC
2 3
= −
‫ا‬2:
ABC‫و‬I‫و‬J‫و‬K‫ث‬
3
BI BC
2
=‫و‬
1
CJ CA
3
=‫و‬
2
AK AB
5
=.
‫أن‬:‫ا‬I‫و‬J‫و‬K.
‫ا‬3:
‫ا‬‫ت‬‫ا‬:
1(u AC BA CB= + −
2(v DE DF EF ED= − + −.
3(w BA MA MB= + −.
‫ا‬4:
A‫و‬B‫و‬M‫و‬I‫أر‬:( )1
MI MA MB
2
= +.
‫أن‬:I‫ا‬[AB].
‫ا‬5:
A‫و‬B‫و‬C‫و‬ ، ‫و‬ ‫ث‬I‫و‬J‫ا‬[AB]‫و‬[BC]‫ا‬ ‫ا‬:
1(‫أن‬:
1
IJ AC
2
=.
2(‫أن‬:IB IC AC+ =.
3(‫ا‬ ‫أ‬M:MA 3MB 2MC 0+ + =
4(‫ا‬ ‫أن‬I‫و‬J‫و‬M.
‫ا‬6:
ABC‫و‬D‫و‬E‫و‬F:
3 2
BF BE ‫و‬ AE 2AD ‫و‬ BD BC
5 3
= = − =
1(‫ا‬ ‫أ‬.
2(‫أن‬:
4
EA 2AB BC
3
= +.
3(‫أن‬:
9 4
FB AB BC
5 5
= +.
4(‫ا‬ ‫أن‬ ‫أ‬A‫و‬F‫و‬C.
5(‫ا‬ ‫أن‬ ‫ا‬(AC)‫و‬(BE)‫ا‬ ‫ن‬F
‫ا‬7:
ABC‫و‬ ،M‫و‬N‫و‬P‫ت‬ ‫ا‬ ‫ا‬ ‫ه‬[BC]‫و‬[AC]‫و‬[AB]:‫أن‬:AM BN CP 0+ + =
‫ا‬8:
ABC،I‫و‬J‫و‬K‫ث‬:
1 3
a ‫و‬ AK aAB ‫و‬ CJ CA ‫و‬ BI BC
3 2
∗∈ = = =»
‫د‬a‫ا‬ ‫أن‬ ‫إذا‬I‫و‬J‫و‬K.

2. http://ad2math.voila.net/ adnanemath@gmail.com ‫ط‬ ‫أآ‬ ‫ن‬ .‫ذ‬ Page 2
‫ا‬9:
ABC،
1(‫ا‬ ‫د‬AB‫ا‬CA ‫و‬ CB.
2(MCM 3MB=‫أآ‬ ،AM‫ا‬CA ‫و‬ CB.
3(‫ا‬ ‫ه‬A‫و‬M‫و‬B‫؟‬‫؟‬ ‫ا‬
‫ا‬10:
ABC‫و‬ ،M‫و‬N‫ن‬ ‫ن‬
3 1
NA NC ‫و‬ AM AB
2 3
= =:
1(‫أن‬MC ‫و‬ BN‫ن‬.
2(I‫ا‬[BN]‫ا‬ ‫ا‬ ،DCI DI 0+ =‫ا‬ ‫أن‬ ،BCND‫ع‬ ‫ا‬ ‫ازي‬.
‫ا‬11:
ABC،B'‫و‬C'( )AC' (1 k)AC ‫و‬ AB' kAB k= − = ∈»،I[B'C'].
1(‫أن‬:
1 k k
AI AC AB
2 2
−
= +.
2(‫ا‬A'I[AA']‫أن‬ ،:( )BA' 1 k BC= −.
3(‫أن‬:( )IA kIB 1 k IC 0+ + − =.
‫ا‬12:)Euler(
ABC‫و‬O‫و‬ ‫ا‬ ‫دا‬ ‫آ‬H‫ا‬ ‫ه‬:OA OB OC OH+ + =.
1(‫ا‬P:OP OA OB= +
a(‫ا‬ ‫ه‬OAPB‫؟‬
b(‫ا‬ ‫ه‬OPHC‫؟‬
c(‫أن‬ ‫ا‬(CH)‫ع‬ ‫ار‬ABC.
2(‫ا‬ ‫أن‬(BH)‫ع‬ ‫ار‬ABC‫ا‬ ‫ا‬ ،HABC.
3(G‫ا‬ ‫آ‬ABC‫ا‬ ‫أن‬ ،O‫و‬H‫و‬G.
):G‫ا‬ ‫آ‬ABC‫أن‬:GA GB GC 0+ + =(
4(‫ا‬ ‫د‬ ‫ا‬ ‫د‬k:OH kOG=.
‫ا‬13:
ABCD‫آ‬ ‫ع‬ ‫أ‬ ‫ازي‬G.
1(E[AB]‫و‬F[BC]‫أن‬ ،:AF BG CE 0+ + =
2(‫ا‬H‫و‬K‫و‬I:
i.DH CE=
ii.K[GH].
iii.I[AD].
a(‫أن‬A‫ه‬[HE].
b(‫ا‬ ‫ه‬AGIH‫؟‬
c(‫أن‬:1
AK AD
4
=
3(
a(‫أن‬DC DB DA 0− + =
b(‫ا‬ ‫د‬M‫ا‬ ‫ن‬MA MB MC− +CE.
4(L‫ا‬:
2
AL AB
3
=‫و‬N[DL]‫و‬[AC]‫أن‬ ،:3NA 2NC 0+ =.