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-2- تمارين الرياضيات للجدع مشترك علوم

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Najlaà Maslouhy
Najlaà Maslouhy

This document contains 13 sections about geometry problems involving points, lines, angles, and relationships between lengths in triangles and quadrilaterals. The problems include determining angle measures, using properties of isosceles and equilateral triangles, applying trigonometric ratios, and utilizing theorems such as the angle sum theorem for triangles and quadrilaterals. The goal is to find unknown values based on given information through logical reasoning and applications of geometric rules.

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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.
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+ =.

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-2- تمارين الرياضيات للجدع مشترك علوم

  • 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+ =.