1. The document contains 11 geometry problems involving lines, angles, and triangles. 2. The problems use basic properties like parallel lines, proportional line segments, and angle bisectors to establish relationships between parts of figures. 3. Equations are set up and solved to find values based on given information in each diagrammatic problem.

1. http://ad2math.voila.net/ adnanemath@gmail.com ‫ط‬ ‫أآ‬ ‫ن‬ .‫ذ‬ Page 1
‫ط‬ ‫ــ‬ ‫ــ‬ ‫ا‬
‫ا‬1:
ABC،‫و‬M:
2
AM= AB
5
‫و‬M1‫ا‬M‫ا‬(AC)‫ا‬ ‫ازي‬(BC).
‫و‬M2‫ا‬M1‫ا‬(BC)‫ا‬ ‫ازي‬(BA).
‫و‬M'‫ا‬M2‫ا‬(AB)‫ا‬ ‫ازي‬(AC).
1(‫أآ‬BM 'BA.
2(‫أن‬ ‫ا‬[AB]‫و‬[MM']‫ا‬.
‫ا‬2:
ABC،E‫و‬F(AB)‫و‬(AC)‫ا‬ ‫ا‬.‫ـ‬ ‫ازي‬ ‫ا‬(CE)‫ر‬ ‫ا‬ ‫و‬F(AB)E'‫و‬
‫ـ‬ ‫ازي‬ ‫ا‬(BF)‫ر‬ ‫ا‬ ‫و‬E(AC)F'.
1(‫أن‬:AE' AC AF' AB× = ×
2(‫أن‬ ‫ا‬:(BC) // (E'F').
‫ا‬3:
ABCD‫ن‬ ‫ا‬ ‫ب‬ ‫ر‬O.‫ر‬ ‫ا‬ ‫ا‬O‫ـ‬ ‫ازي‬ ‫ا‬ ‫و‬(BC)(AB)E‫و‬ ،
‫ـ‬ ‫ازي‬ ‫ا‬(DC)‫ـ‬ ‫ر‬ ‫ا‬ ‫و‬O‫ا‬(AD)F.
‫أن‬:(BD) // (EF).
‫ا‬4:
ABCD‫ب‬ ‫ر‬‫و‬M
1
BM BA
3
=.
N‫ا‬M(BC)‫ازي‬(AC).
‫و‬P‫ا‬N(CD)‫ازي‬(BD).
1(‫أن‬:
1
DP DC
3
=.
2(‫ا‬Q
1
DQ DA
3
=‫ا‬ ‫أن‬ ،MNPQ‫ع‬ ‫ا‬ ‫ازي‬.
‫ا‬5:
ABC‫و‬M‫ا‬[AB]‫و‬M'(AC)‫ازي‬(BC).‫ا‬D‫ا‬ ‫ه‬M'
(BC)‫ازي‬(AB):
‫أن‬:
MM' CD
1
BC CB
= −
‫ا‬6:
ABCD‫ف‬
10
DC AB
3
=‫و‬I‫و‬J
4
JA JD
3
=‫و‬
4
IA ID
3
−
=.
‫ـ‬ ‫ن‬ ‫از‬ ‫ا‬(AB)‫ر‬ ‫ا‬ ‫و‬I‫و‬J‫ن‬(BC)N‫و‬Q‫ا‬ ‫ا‬.‫ـ‬ ‫ازي‬ ‫ا‬(AD)‫ر‬ ‫ا‬ ‫و‬B(DC)
E‫و‬(NI)K‫و‬(JQ)H.
‫اد‬ ‫ا‬ ‫د‬a‫و‬b‫و‬c:HQ b.AB ‫و‬ EC c.AB ‫و‬ KN a.CE= = =.
‫ا‬7:‫ن‬ ‫ه‬ ‫ا‬ ‫ه‬ ‫ا‬ ‫ا‬ ‫ه‬ ‫ف‬ ‫ا‬‫ه‬CEVA:
ABC‫و‬ ‫م‬M‫و‬N‫و‬P‫ا‬ ‫ا‬ ‫ث‬(BC)‫و‬(AC)‫و‬(AB)‫ـ‬ ‫و‬A‫و‬B‫و‬C.
:(AM) // (BN) // (CP).
1(‫أن‬:
AB MB
AP MC
=.
2(‫أن‬:
BP NC
BA NA
=.

2. http://ad2math.voila.net/ adnanemath@gmail.com ‫ط‬ ‫أآ‬ ‫ن‬ .‫ذ‬ Page 2
3(‫أن‬:
MB NC PA
1
MC NA PB
× × =.
‫ا‬8:
ABC‫و‬ ‫م‬M‫و‬N‫و‬P‫ا‬ ‫ا‬ ‫ا‬ ‫ث‬[BC]‫وا‬(AC)‫ا‬ ‫و‬(AB)‫ـ‬ ‫و‬A
‫و‬B‫و‬C.:(AM) // (BN) // (CP).)‫ا‬ ‫أ‬(
1(‫أن‬:
MA CM MA BM
‫و‬
BN CB CP BC
= =
2(‫أن‬ ‫ا‬:
1 1 1
AM BN CP
= +
3(‫أ‬ ‫ا‬ ‫ا‬ ‫ل‬h‫ا‬a‫و‬b
‫أن‬:
1 1 1
h a b
= +.
‫ا‬9:
ABC،‫و‬G‫آ‬.
‫ا‬(D)G‫ت‬ ‫ا‬ ‫و‬(AG)‫و‬(BG)‫و‬(CG)‫ا‬ ‫ا‬M‫و‬N‫و‬P.
:GC zGP ‫و‬ GB yGN ‫و‬ GA xGM= = =x‫و‬y‫و‬z‫اد‬ ‫أ‬.
‫ل‬p‫ا‬ ‫ط‬ ‫ا‬(AG)‫ا‬ ‫ازي‬(D).
‫أن‬:x y z 0+ + =.
‫ا‬10:)‫ه‬‫س‬MENELAUS(
ABC‫و‬(D)‫ا‬ ‫رؤوس‬ ‫رأس‬ ‫أي‬ABC‫و‬(BC)‫و‬(CA)‫و‬(AB)
‫ا‬ ‫ا‬M‫و‬N‫و‬P.
‫أن‬ ‫أ‬:
MB NC PA
1
MC NA PB
× × =
‫ا‬11:
ABC‫و‬M{ }1 ‫و‬ AM .AB∗α∈ − = α».N‫ا‬M(AC)‫ازي‬
(BC)‫و‬H‫دي‬ ‫ا‬ ‫ا‬A(BC).
I(MN)‫و‬(AH).
1(‫أن‬:AI .AH ‫و‬ MN .BC= α = α
2(‫أن‬:
S
²
S'
= αS‫ا‬AMN‫و‬S'‫ا‬ABC.