Rota

  • December 2019
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‫اﻟﺪوران‬ ‫‪ -I‬ﺗﻌﺮﻳﻒ اﻟﺪوران‬ ‫‪ -1‬ﺗﻌﺮﻳﻒ‬ ‫ﻟﺘﻜﻦ ‪ O‬ﻧﻘﻄﺔ ﻣﻦ اﻟﻤﺴﺘﻮى اﻟﻤﻮﺟﻪ ‪ P‬و ‪ α‬ﻋﺪدا ﺣﻘﻴﻘﻴﺎ‬ ‫اﻟﺪوران اﻟﺬي ﻣﺮآﺰﻩ ‪ O‬و زاوﻳﺘﻪ ‪ α‬هﻮ اﻟﺘﻄﺒﻴﻖ ﻣﻦ ‪ P‬ﻧﺤﻮ ‪ P‬اﻟﺬي ﻳﺮﺏﻂ آﻞ ﻧﻘﻄﺔ ‪ M‬ﺏﻨﻘﻄﺔ ' ‪ M‬ﺏﺤﻴﺚ‪:‬‬ ‫‪ M ' = O -‬إذا آﺎﻧﺖ ‪M = O‬‬

‫‪-‬‬

‫] ‪[ 2π‬‬

‫' ‪ OM = OM‬‬ ‫‪‬‬ ‫‪  JJJJG JJJJJG‬إذا آﺎن ‪M ≠ O‬‬ ‫‪ OM ; OM ' ≡ α‬‬

‫)‬

‫(‬

‫*‪ -‬ﻧﺮﻣﺰ ﻟﻠﺪوران اﻟﺬي ﻣﺮآﺰﻩ ‪ O‬و زاوﻳﺘﻪ ‪ α‬ﺏﺎﻟﺮﻣﺰ ) ‪ r ( O;α‬أو ﺏﺎﻟﺮﻣﺰ ‪r‬‬ ‫*‪ -‬اﻟﻨﻘﻄﺔ ' ‪ M‬ﺗﺴﻤﻰ ﺻﻮرة ‪ M‬ﺏﺎﻟﺪوران ‪ r‬ﻧﻜﺘﺐ ' ‪r ( M ) = M‬‬

‫ﻣﺜﺎل‬ ‫‪π‬‬

‫ﻟﺘﻜﻦ ‪ O‬و ‪ A‬و ‪ B‬ﺙﻼث ﻧﻘﻂ و ‪ r‬اﻟﺪوران اﻟﺬي ﻣﺮآﺰﻩ ‪ O‬و زاوﻳﺘﻪ‬

‫‪6‬‬

‫أﻧﺸﺊ ' ‪ A‬و ' ‪ B‬ﺻﻮرﺗﻲ ‪ A‬و ‪ B‬ﻋﻠﻰ اﻟﺘﻮاﻟﻲ ﺏﺎﻟﺪوران ‪r‬‬

‫‪ – 2‬اﺱﺘﻨﺘﺎﺟﺎت‬ ‫أ( اﻟﻤﺜﻠﺚ اﻟﻤﺘﺴﺎوي اﻟﺴﺎﻗﻴﻦ‬

‫(‬

‫)‬

‫‪JJJ‬‬ ‫‪G JJJG‬‬ ‫‪n‬‬ ‫‪ ABC -‬ﻣﺜﻠﺚ ﻣﺘﺴﺎوي اﻟﺴﺎﻗﻴﻦ رأﺱﻪ ‪ A‬ﻳﻌﻨﻲ أن اﻟﺪوران اﻟﺬي ﻣﺮآﺰﻩ ‪ A‬و زاوﻳﺘﻪ ‪ AB; AC‬ﻳﺤﻮل ‪ B‬إﻟﻰ ‪C‬‬

‫‪-‬‬

‫إذا آﺎن ‪ ABC‬ﻣﺜﻠﺚ ﻣﺘﺴﺎوي اﻟﺴﺎﻗﻴﻦ وﻗﺎﺋﻢ اﻟﺰاوﻳﺔ ﻓﻲ ‪A‬‬

‫ﻣﺮآﺰﻩ ‪ A‬و زاوﻳﺘﻪ‬ ‫‪-‬‬

‫‪π‬‬ ‫‪2‬‬

‫زاوﻳﺘﻪ‬

‫‪3‬‬

‫ﺏﺤﻴﺚ ] ‪[ 2π‬‬

‫ﻳﺤﻮل ‪ B‬إﻟﻰ ‪C‬‬

‫ب( اﻟﺪوران اﻟﺬي زاوﻳﺘﻪ ﻣﻨﻌﺪﻣﺔ‬ ‫ﻟﻴﻜﻦ ) ‪ r ( O;α‬دوراﻧﺎ‬

‫‪ -‬إذا آﺎن ] ‪[ 2π‬‬

‫‪2‬‬

‫(‬

‫ﻳﺤﻮل ‪ B‬إﻟﻰ ‪C‬‬

‫ إذا آﺎن ‪ ABC‬ﻣﺜﻠﺚ ﻣﺘﺴﺎوي اﻷﺿﻼع‬‫‪π‬‬

‫ﺏﺤﻴﺚ ] ‪[ 2π‬‬

‫‪π‬‬

‫)‬

‫‪JJJ‬‬ ‫‪G JJJG‬‬ ‫‪n‬‬

‫≡ ‪ AB; AC‬ﻓﺎن اﻟﺪوران اﻟﺬي‬

‫‪ α ≡ 0‬ﻓﺎن ‪r ( M ) = M‬‬

‫ﻓﻲ هﺬﻩ اﻟﺤﺎﻟﺔ ‪ r‬هﻮ اﻟﺘﻄﺒﻴﻖ اﻟﻤﺘﻄﺎﺏﻖ ﻓﻲ اﻟﻤﺴﺘﻮى‬

‫‪π‬‬ ‫‪3‬‬

‫)‬

‫‪JJJ‬‬ ‫‪G JJJG‬‬ ‫‪n‬‬

‫(‬

‫≡ ‪ AB; AC‬ﻓﺎن اﻟﺪوران اﻟﺬي ﻣﺮآﺰﻩ ‪ A‬و‬

‫ﺟﻤﻴﻊ ﻧﻘﻂ اﻟﻤﺴﺘﻮى ﺻﺎﻣﺪة‬

‫إذا آﺎن ] ‪[ 2π‬‬

‫‪ α ≡/ 0‬ﻓﺎن اﻟﻨﻘﻄﺔ اﻟﻮﺣﻴﺪ اﻟﺼﺎﻣﺪة ﺏﺎﻟﺪوران ‪ r‬هﻲ ﻣﺮآﺰﻩ ‪O‬‬

‫ج( اﻟﺪوران اﻟﺬي زاوﻳﺘﻪ ﻣﺴﺘﻘﻴﻤﻴﺔ‬ ‫‪r ( O; π ) = S O‬‬

‫‪ -3‬اﻟﺪوران اﻟﻌﻜﺴﻲ‬ ‫ﻟﻴﻜﻦ ) ‪ r ( O;α‬دوراﻧﺎ‬ ‫' ‪ OM = OM‬‬ ‫‪‬‬ ‫‪r ( M ) = M ' ⇔  JJJJG JJJJJG‬‬ ‫] ‪ OM ; OM ' ≡ α [ 2π‬‬ ‫‪ OM ' = OM‬‬ ‫‪‬‬ ‫‪r ( M ) = M ' ⇔  JJJJJG JJJJG‬‬ ‫] ‪ OM '; OM ≡ −α [ 2π‬‬ ‫) ‪r ( M ) = M ' ⇔ r ' ( M ') = M / r ' = r ( O; −α‬‬

‫)‬

‫(‬

‫)‬

‫(‬

‫اﻟﺪوران ) ‪ r ( O; −α‬ﻳﺴﻤﻰ اﻟﺪوران اﻟﻌﻜﺴﻲ ﻟﻠﺪوران ) ‪ r ( O;α‬ﻧﺮﻣﺰ ﻟﻪ ﺏﺎﻟﺮﻣﺰ ‪r −1‬‬ ‫‪−1‬‬ ‫' ‪r ( M ) = M‬‬ ‫‪r ( M ') = M‬‬ ‫‪⇔‬‬ ‫‪ −1‬‬ ‫‪ r (O ) = O‬‬ ‫‪ r ( O ) = O‬‬

‫اﻟﺪوران ‪ r‬ﺗﻄﺒﻴﻖ ﺗﻘﺎﺏﻠﻲ ﻓﻲ اﻟﻤﺴﺘﻮى‬ ‫ﺧﺎﺻﻴﺔ‬ ‫آﻞ دوران ) ‪ r ( O;α‬هﻮ ﺗﻄﺒﻴﻖ ﺗﻘﺎﺏﻠﻲ ﻓﻲ اﻟﻤﺴﺘﻮى‬ ‫اﻟﺪوران ) ‪ r ( O; −α‬ﻳﺴﻤﻰ اﻟﺪورا ن اﻟﻌﻜﺴﻲ ﻟﻠﺪوران ) ‪ r ( O;α‬ﻧﺮﻣﺰ ﻟﻪ ﺏـ‪r −1 :‬‬

‫ﺗﻤﺎرﻳﻦ ﺗﻄﺒﻴﻘﻴﺔ‬ ‫‪ -1‬ﻟﻴﻜﻦ ‪ ABCD‬ﻣﺮﺏﻌﺎ‬ ‫ﺣﺪد زاوﻳﺘﻲ اﻟﺪوارﻧﻴﻴﻦ ‪ r1‬و ‪ r2‬اﻟﺬي ﻣﺮآﺰاهﻤﺎ ‪ A‬و ‪ C‬ﻋﻠﻰ اﻟﺘﻮاﻟﻲ وﻳﺤﻮﻻن ﻣﻌﺎ اﻟﻨﻘﻄﺔ ‪ D‬إﻟﻰ ‪B‬‬

‫‪ -2‬ﻟﻴﻜﻦ ‪ ABC‬ﻣﺜﻠﺚ ﻣﺘﺴﺎوي اﻷﺿﻼع ﺣﻴﺚ‬ ‫أ‪ -‬ﺣﺪد ﻣﺮآﺰ اﻟﺪوران ‪ r‬اﻟﺬي ﻳﺤﻮل ‪ B‬إﻟﻰ ‪C‬‬

‫ب‪ -‬ﺣﺪد اﻟﺪوران اﻟﻌﻜﺴﻲ ﻟﻠﺪوران ‪r‬‬

‫‪ -II‬ﺧﺎﺻﻴﺎت اﻟﺪوران‬ ‫‪ -1‬ﺧﺎﺻﻴﺔ أﺱﺎﺱﻴﺔ ) اﻟﺤﻔﺎظ ﻋﻠﻰ اﻟﻤﺴﺎﻓﺔ(‬ ‫ﻟﻴﻜﻦ ) ‪ r ( O;α‬دوراﻧﺎ و ‪ A‬و ‪ B‬ﻧﻘﻄﺘﻴﻦ‬

‫] ‪[ 2π‬‬

‫‪n π‬‬ ‫≡ ) ‪; CB‬‬ ‫‪(CA‬‬ ‫‪3‬‬ ‫‪JJJG JJJG‬‬

r ( B ) = B ' ; r ( A) = A '

AB = A ' B ' ‫ﻟﻨﻘﺎرن‬

:‫ ﻟﺪﻳﻨﺎ‬OA ' B ' ‫ و‬OAB ‫ﺣﺴﺐ ﻋﻼﻗﺔ اﻟﻜﺎﺵﻲ ﻓﻲ اﻟﻤﺜﻠﺜﻴﻦ‬ AB 2 = OA2 + OB 2 − 2OA ⋅ OB.cos  n AOB    AB '2 = OA '2 + OB '2 − 2OA '⋅ OB '.cos  n A ' OB '  

 OB = OB '  JJJG JJJJG   OB; OB ' ≡ α

(

)

[ 2π ]

‫و‬

 OA = OA '  JJJG JJJG   OA; OA ' ≡ α

(

)

[ 2π ]

:‫ ﻓﺎن‬r ( B ) = B ' ; r ( A ) = A ' ‫و ﺏﻤﺎ أن‬ ‫و ﻟﺪﻳﻨﺎ ﻣﻦ ﺟﻬﺔ أﺧﺮى‬

) ( ) ( ) ( ( JJJG JJJG JJJG JJJJG O A O B O ; ≡ + α ) ( A '; O B ' ) − α [2π ] ( JJJG JJJG JJJG JJJJG O A O B O ; ≡ ) ( A '; O B ' ) [2π ] (

JJJG JJJG JJJG JJJG JJJG JJJJG JJJJG JJJG O A ; O B ≡ O A ; O A ' + O A '; O B ' + O B '; O B

)

[ 2π ]

n AOB  =  n A ' OB ' ‫وﻣﻨﻪ‬     A ' B '2 = OA2 + OB 2 − 2OA ⋅ OB.cos  n AOB  ‫و ﺏﺎﻟﺘﺎﻟﻲ‬  

A ' B ' = AB ‫ اذن‬A ' B '2 = AB 2 ‫وﻣﻨﻪ‬

‫ﺧﺎﺻﻴﺔ‬ ‫ ﻧﻘﻄﺘﻴﻦ‬B ‫ و‬A ‫ دوراﻧﺎ و و‬r ‫ﻟﻴﻜﻦ‬ A ' B ' = AB ‫ ﻓﺎن‬r ( B ) = B ' ; r ( A ) = A ' ‫إذا آﺎن‬

‫ﻧﻌﺒﺮ ﻋﻦ هﺬا ﺏﻘﻮﻟﻨﺎ اﻟﺪوران ﻳﺤﺎﻓﻆ ﻋﻠﻰ اﻟﻤﺴﺎﻓﺔ‬ ‫ﺗﻤﺮﻳﻦ‬ ‫ ﻣﺜﻠﺜﺎن ﻣﺘﺴﺎوﻳﺎ اﻷﺿﻼع‬NAC ‫ و‬MAB ‫ ﻧﻘﻄﺘﻴﻦ ﺧﺎرج اﻟﻤﺴﺘﻮى ﺏﺤﻴﺚ‬N ‫ و‬M ‫ ﻧﻌﺘﺒﺮ‬. ‫ ﻣﺜﻠﺜﺎ‬ABC ‫ﻟﻴﻜﻦ‬ NB ‫ و‬MC ‫ﻗﺎرن‬

‫ اﻟﺪوران و اﺱﺘﻘﺎﻣﻴﺔ اﻟﻨﻘﻂ‬-2 ‫أ( ﺻﻮرة ﻗﻄﻌﺔ‬ r ‫ ﺏﺪوران‬B ‫ و‬A ‫ ﺻﻮرﺗﻲ‬B ' ‫ و‬A ' ‫ [ ﻗﻄﻌﺔ و‬AB ] ‫ﻟﺘﻜﻦ‬

r ‫ ﺻﺮﺗﻬﺎ ﺏﺎﻟﺪوران‬M ' ‫ [ و‬AB ] ‫ ﻧﻘﻄﺔ ﻣﻦ‬M ‫ﻟﺘﻜﻦ‬ M ' ∈ [ A ' B '] ‫ ﺏﻴﻦ أن‬-1

‫‪JJJJG‬‬

‫‪JJJG‬‬

‫‪ -2‬ﺏﻴﻦ اذا آﺎن ‪ AM = λ AB‬ﺣﻴﺚ ‪ λ 0 ≤ λ ≤ 1‬ﻓﺎن‬

‫‪JJJJJG‬‬ ‫‪JJJJJG‬‬ ‫' ‪AM ' = λ A ' B‬‬

‫ﺧﺎﺻﻴﺔ‬ ‫ﻟﺘﻜﻦ ] ‪ [ AB‬ﻗﻄﻌﺔ و ' ‪ A‬و ' ‪ B‬ﺻﻮرﺗﻲ ‪ A‬و ‪ B‬ﺏﺪوران ‪r‬‬

‫ﺻﻮرة اﻟﻘﻄﻌﺔ ] ‪ [ AB‬ﺏﺎﻟﺪوران‬ ‫‪JJJG‬‬

‫‪ r‬هﻲ اﻟﻘﻄﻌﺔ ]' ‪[ A ' B‬‬

‫‪JJJJG‬‬

‫اذا آﺎن ‪ AM = λ AB‬ﺣﻴﺚ ‪ λ 0 ≤ λ ≤ 1‬ﻓﺎن‬

‫‪JJJJJG‬‬ ‫‪JJJJJG‬‬ ‫' ‪ AM ' = λ A ' B‬ﺣﻴﺚ ' ‪r ( M ) = M‬‬

‫ب‪ -‬ﺻﺮة ﻣﺴﺘﻘﻴﻢ‬ ‫ﻟﺘﻜﻦ ' ‪ A‬و ' ‪ B‬ﺻﻮرﺗﻲ اﻟﻨﻘﻄﺘﻴﻦ اﻟﻤﺨﺘﻠﻔﺘﻴﻦ ‪ A‬و ‪ B‬ﺏﺪوران ‪r‬‬ ‫أ‪ -‬ﺏﻴﻦ أن )' ‪r ([ AB ) ) = [ A ' B‬‬

‫ب‪ -‬ﺏﻴﻦ أن )' ‪r ( ( AB ) ) = ( A ' B‬‬

‫ﺧﺎﺻﻴﺔ‬ ‫ﻟﺘﻜﻦ ' ‪ A‬و ' ‪ B‬ﺻﻮرﺗﻲ ﻧﻘﻄﺘﻴﻦ ﻣﺨﺘﻠﻔﺘﻴﻦ ‪ A‬و ‪ B‬ﻋﻠﻰ اﻟﺘﻮاﻟﻲ ﺏﺪوران ‪r‬‬

‫* ﺻﻮرة ﻧﺼﻒ اﻟﻤﺴﺘﻘﻴﻢ ) ‪ [ AB‬هﻮ ﻧﺼﻒ اﻟﻤﺴﺘﻘﻴﻢ‬

‫)' ‪[ A ' B‬‬

‫* ﺻﻮرة اﻟﻤﺴﺘﻘﻴﻢ ) ‪ ( AB‬هﻮ اﻟﻤﺴﺘﻘﻴﻢ )' ‪( A ' B‬‬ ‫‪JJJG‬‬

‫‪JJJJG‬‬

‫* إذا آﺎن ‪ AM = λ AB‬ﺣﻴﺚ \ ∈ ‪ λ λ‬ﻓﺎن‬

‫‪JJJJJG‬‬ ‫‪JJJJJG‬‬ ‫' ‪ AM ' = λ A ' B‬ﺣﻴﺚ ' ‪r ( M ) = M‬‬

‫ج‪ -‬اﻟﻤﺮﺟﺢ و اﻟﺪوران‬ ‫' ‪ A‬و ' ‪ B‬و ' ‪ G‬ﺻﻮر اﻟﻨﻘﻂ ‪ A‬و ‪B‬‬

‫و ‪ G‬ﺏﺪوران ‪ r‬ﻋﻠﻰ اﻟﺘﻮاﻟﻲ و ‪ G‬ﻣﺮﺟﺢ ) ‪ ( A;α‬و ) ‪( B; β‬‬

‫ﺏﻴﻦ أن ' ‪ G‬ﻣﺮﺟﺢ ) ‪ ( A ';α‬و ) ‪( B '; β‬‬ ‫ﺧﺎﺻﻴﺔ‬ ‫' ‪ A‬و ' ‪ B‬و ' ‪ G‬ﺻﻮر اﻟﻨﻘﻂ ‪ A‬و ‪ B‬و ‪ G‬ﺏﺪوران ‪ r‬ﻋﻠﻰ اﻟﺘﻮاﻟﻲ‬ ‫إذا آﺎن ‪ G‬ﻣﺮﺟﺢ ) ‪ ( A;α‬و ) ‪ ( B; β‬ﻓﺎن‬

‫' ‪ G‬ﻣﺮﺟﺢ ) ‪ ( A ';α‬و ) ‪( B '; β‬‬

‫اﻟﺪوران ﻳﺤﺎﻓﻆ ﻋﻠﻰ ﻣﺮﺟﺢ ﻧﻘﻄﺘﻴﻦ‬ ‫ﻣﻼﺣﻈﺔ‪ :‬اﻟﺨﺎﺻﻴﺔ ﺗﺒﻘﻰ ﺻﺤﻴﺤﺔ ﻟﻤﺮﺟﺢ أآﺜﺮ ﻣﻦ ﻧﻘﻄﺘﻴﻦ‬ ‫ﻧﺘﻴﺠﺔ‬ ‫' ‪ A‬و ' ‪ B‬و ' ‪ I‬ﺻﻮراﻟﻨﻘﻂ ‪ A‬و ‪ B‬و ‪ I‬ﺏﺪوران ‪ r‬ﻋﻠﻰ اﻟﺘﻮاﻟﻲ‬ ‫إذا آﺎن ‪ I‬ﻣﻨﺘﺼﻒ ] ‪ [ AB‬ﻓﺎن ' ‪I‬‬

‫اﻟﺪوران ﻳﺤﺎﻓﻆ ﻋﻠﻰ اﻟﻤﻨﺘﺼﻒ‬

‫ﻣﻨﺘﺼﻒ ]' ‪[ A ' B‬‬

‫د( اﻟﺤﻔﺎظ ﻋﻠﻰ ﻣﻌﺎﻣﻞ اﻻﺱﺘﻘﺎﻣﻴﺔ‬ ‫ﺧﺎﺻﻴﺔ ) ﻣﻘﺒﻮﻟﺔ(‬ ‫ﻟﺘﻜﻦ ' ‪ A‬و ' ‪ B‬و '‪ C‬و ' ‪ D‬ﺻﻮر أرﺏﻊ ﻧﻘﻂ ‪ A‬و ‪ B‬و ‪ C‬و ‪ D‬ﺏﺪوران ‪ r‬ﻋﻠﻰ اﻟﺘﻮاﻟﻲ و \ ∈ ‪λ‬‬ ‫‪JJJJJG‬‬ ‫‪JJJJJG‬‬ ‫‪JJJG‬‬ ‫‪JJJG‬‬ ‫إذا آﺎن ‪ CD = λ AB‬ﻓﺎن ' ‪C ' D ' = λ A ' B‬‬

‫ﻧﻌﺒﺮ ﻋﻦ هﺬا ﺏﻘﻮﻟﻨﺎ اﻟﺪوران ﻳﺤﺎﻓﻆ ﻋﻠﻰ ﻣﻌﺎﻣﻞ اﺱﺘﻘﺎﻣﻴﺔ ﻣﺘﺠﻬﺘﻴﻦ‬ ‫ﺗﻤﺮﻳﻦ‬ ‫ﻟﻴﻜﻦ ‪ ABCD‬ﻣﺮﺏﻌﺎ‬ ‫ﻧﻨﺸﺊ ﺧﺎرﺟﻪ اﻟﻤﺜﻠﺚ ‪ CBF‬اﻟﻤﺘﺴﺎوي اﻷﺿﻼع و داﺧﻠﻪ اﻟﻤﺜﻠﺚ ‪ABE‬‬

‫‪π‬‬ ‫‪‬‬ ‫ﺏﻴﻦ أن اﻟﻨﻘﻂ ‪ D‬و ‪ E‬و ‪ F‬ﻣﺴﺘﻘﻴﻤﻴﺔ ) ﻳﻤﻜﻦ اﻋﺘﺒﺎر اﻟﺪورن ‪r = r  B; − ‬‬ ‫‪3‬‬ ‫‪‬‬

‫‪ -3‬اﻟﺪوران و اﻟﺰواﻳﺎ‬ ‫أ( ﺧﺎﺻﻴﺔ أﺱﺎﺱﻴﺔ‬

‫‪JJJG JJJG‬‬ ‫ﻟﺘﻜﻦ ' ‪ A‬و ' ‪ B‬ﺻﻮرﺗﻲ ‪ A‬و ‪ B‬ﺏﺪوران ‪ r‬زاوﻳﺘﻪ ‪ α‬ﻋﻠﻰ اﻟﺘﻮاﻟﻲ ‪ .‬ﻟﺘﻜﻦ ‪ C‬ﻧﻘﻄﺔ ﺣﻴﺚ ‪OC = AB‬‬ ‫‪JJJG JJJJJG‬‬ ‫ﻟﺘﻜﻦ ' ‪ r ( C ) = C‬وﻣﻨﻪ ' ‪OC ' = A ' B‬‬

‫و ﺏﺎﻟﺘﺎﻟﻲ ] ‪[ 2π‬‬ ‫وﺣﻴﺚ أن ] ‪[ 2π‬‬

‫‪n‬‬ ‫‪n‬‬ ‫)' ‪; OC ') ≡ ( AB; A ' B‬‬ ‫‪(OC‬‬ ‫‪JJJG‬‬ ‫‪JJJJG‬‬ ‫‪n‬‬ ‫‪; OC ') ≡ α‬‬ ‫‪ ( OC‬ﻓﺎن ] ‪[ 2π‬‬ ‫‪JJJG JJJJJG‬‬

‫‪JJJG JJJJG‬‬

‫‪AB; A ' B ') ≡ α‬‬ ‫‪(n‬‬ ‫‪JJJG JJJJJG‬‬

‫ﺧﺎﺻﻴﺔ‬ ‫ﻟﻴﻜﻦ ‪ r‬دواراﻧﺎ زاوﻳﺘﻪ ‪α‬‬

‫إذا آﺎن ' ‪ A‬و ' ‪ B‬ﺻﻮرﺗﻲ ‪ A‬و ‪ B‬ﺏﺎﻟﺪوران ‪ r‬ﻓﺎن‬

‫] ‪[ 2π‬‬

‫)‬

‫‪JJJ‬‬ ‫‪G JJJJJG‬‬ ‫‪n‬‬ ‫‪AB; A ' B ' ≡ α‬‬

‫ب‪ -‬ﻧﺘﻴﺠﺔ‬

‫(‬ ‫( )‬ ‫( )‬ ‫( )‬ ‫] ‪) [ 2π‬‬ ‫‪JJJ‬‬ ‫‪G JJJG‬‬ ‫‪JJJJJ‬‬ ‫‪G JJJJJG‬‬ ‫‪n‬‬ ‫‪n‬‬ ‫‪A‬‬ ‫‪B‬‬ ‫;‬ ‫‪C‬‬ ‫‪D‬‬ ‫≡‬ ‫‪+‬‬ ‫‪A‬‬ ‫'‬ ‫‪B‬‬ ‫‪α‬‬ ‫(‬ ‫] ‪) ( '; C ' D ') − α [2π‬‬ ‫‪JJJ‬‬ ‫‪G JJJG‬‬ ‫‪JJJJJ‬‬ ‫‪G JJJJJG‬‬ ‫‪n‬‬ ‫‪n‬‬ ‫‪AB‬‬ ‫;‬ ‫‪CD‬‬ ‫≡‬ ‫‪A‬‬ ‫'‬ ‫‪B‬‬ ‫إذن ] ‪( ) ( ';C ' D ') [2π‬‬ ‫‪JJJ‬‬ ‫‪G JJJG‬‬ ‫‪JJJ‬‬ ‫‪G JJJJJG‬‬ ‫‪JJJJJ‬‬ ‫‪G JJJJJG‬‬ ‫‪JJJJJ‬‬ ‫‪G JJJG‬‬ ‫‪n‬‬ ‫‪n‬‬ ‫‪n‬‬ ‫‪n‬‬ ‫‪A B ; C D ≡ A B ; A ' B ' + A ' B '; C ' D ' + C ' D '; C D‬‬

‫(‬

‫ﻧﺘﻴﺠﺔ‬ ‫ﻟﺘﻜﻦ ' ‪ A‬و ' ‪ B‬و '‪ C‬و ' ‪ D‬ﺻﻮر أرﺏﻊ ﻧﻘﻂ ‪ A‬و ‪ B‬و ‪ C‬و ‪ D‬ﺏﺪوران ‪ r‬ﺣﻴﺚ ‪ A ≠ B‬و ‪C ≠ D‬‬

‫] ‪[ 2π‬‬

‫)‬

‫( )‬

‫‪JJJ‬‬ ‫‪G JJJG‬‬ ‫‪JJJJJ‬‬ ‫‪G JJJJJG‬‬ ‫‪n‬‬ ‫‪n‬‬ ‫' ‪AB; CD ≡ A ' B '; C ' D‬‬

‫(‬

‫ﻧﻌﺒﺮ ﻋﻦ هﺬا ﺏﻘﻮﻟﻨﺎ اﻟﺪوران ﻳﺤﺎﻓﻆ ﻋﻠﻰ ﻗﻴﺎ س اﻟﺰواﻳﺎ‬ ‫ﺗﻤﺮﻳﻦ‬ ‫‪  p‬اﻟﺬي ﻻ‬ ‫ﻟﻴﻜﻦ ‪ ABC‬ﻣﺜﻠﺜﺎ ﻣﺘﺴﺎوي اﻟﺴﺎﻗﻴﻦ رأﺱﻪ ‪ A‬و ) ‪ ( C‬داﺋﺮة ﻣﺤﻴﻄﺔ ﺏﻪ ‪ .‬ﻧﻌﺘﺒﺮ ‪ M‬ﻧﻘﻄﺔ ﻣﻦ اﻟﻘﻮس ‪AB ‬‬ ‫‪‬‬

‫)‬

‫‪JJJG JJJG‬‬

‫(‬

‫ﻳﺤﺘﻮي ﻋﻠﻰ ‪ . C‬ﻟﻴﻜﻦ ‪ r‬اﻟﺪوران اﻟﺬي ﻣﺮآﺰﻩ ‪ A‬و زاوﻳﺘﻪ ‪. AB; AC‬‬ ‫ﺏﻴﻦ أن ‪ M‬و ' ‪ M‬و ‪ C‬ﻧﻘﻂ ﻣﺴﺘﻘﻴﻤﻴﺔ ﺣﻴﺚ ' ‪r ( M ) = M‬‬

‫‪ -4‬ﺻﻮرة داﺋﺮة ﺏﺪوران‬ ‫ﺧﺎﺻﻴﺔ‬ ‫ﺻﻮرة داﺋﺮة ) ‪ C ( Ω; R‬ﺏﺪوران ‪ r‬هﻲ داﺋﺮة ) ‪ C ( Ω '; R‬ﺣﻴﺚ ' ‪r ( Ω ) = Ω‬‬

‫ﺗﻤﺮﻳﻦ‬ ‫ﻟﻴﻜﻦ ‪ ABCD‬ﻣﺮﺏﻌﺎ و ) ‪ ( C‬داﺋﺮة ﻣﺎرة ﻣﻦ ‪ A‬و ‪C‬‬

‫ﻟﺘﻜﻦ ‪ Q‬و ‪ R‬ﻧﻘﻄﺘﺎ ﺗﻘﺎﻃﻊ ) ‪ ( C‬ﻣﻊ ) ‪ ( BC‬و ) ‪ ( CD‬ﻋﻠﻰ اﻟﺘﻮاﻟﻲ‬ ‫ﺏﻴﻦ أن ‪BQ = DR‬‬

‫) ﻳﻤﻜﻦ اﻋﺘﺒﺎر اﻟﺪوران ‪ r‬اﻟﺬي ﻣﺮآﺰﻩ ‪ A‬و زاوﻳﺘﻪ‬

‫‪π‬‬ ‫‪2‬‬

‫(‬

‫‪‬‬

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