من طرف *ﺣﻘﺎﺋﻖ ﻋﻦ ﺍﻟﻤﺜﻠﺜﺎﺕ*
*ﺗﺸﺎﺑﻪ ﻣﺜﻠﺜﻴﻦ*
*ﻳﻘﺎﻝ ﻋﻦ ﻣﺜﻠﺜﻴﻦ ﺃﻧﻬﻤﺎ ﻣﺘﺸﺎﺑﻬﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﺘﻘﺎﺑﻠﺔ ﻣﻦ ﻛﻞ ﻣﻨﻬﻤﺎ ﻣﺘﺴﺎﻭﻳﺔ، ﺃﻱ ﻋﻨﺪﻣﺎ ﻳﻨﺘﺞ ﺃﺣﺪﻫﻤﺎ ﻋﻦ ﺍﻵﺧﺮ ﺑﺘﻜﺒﻴﺮﻩ ﺃﻭ ﺗﺼﻐﻴﺮﻩ. ﻭﺗﻜﻮﻥ ﺃﻃﻮﺍﻝ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺜﻴﻦ ﺍﻟﻤﺘﺸﺎﺑﻬﻴﻦ ﻣﺘﻨﺎﺳﺒﺔ، ﺃﻱ ﺃﻧﻪ ﺇﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﺃﻗﺼﺮ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﺍﻷﻭﻝ ﻫﻮ ﺿﻌﻔﺎ ﻃﻮﻝ ﺃﻗﺼﺮ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﺎﻧﻲ، ﻓﺈﻥ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻷﻃﻮﻝ ﻭﺍﻟﻤﺘﻮﺳﻂ ﻣﻦ ﺍﻟﻤﺜﻠﺚ ﺍﻷﻭﻝ ﻫﻮ ﺿﻌﻔﺎ ﻃﻮﻟﻲ ﻟﻀﻠﻌﻴﻦ ﺍﻷﻃﻮﻝ ﻭﺍﻟﻤﺘﻮﺳﻂ ﻣﻦ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﺎﻧﻲ ﺃﻳﻀﺎ، ﻭﺑﺎﻟﺘﺎﻟﻲ ﻓﺎﻥ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻃﻮﻟﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻷﻗﺼﺮ ﻭﺍﻷﻃﻮﻝ ﻓﻲ ﺍﻟﻤﺜﻠﺚ ﺍﻷﻭﻝ ﻣﺴﺎﻭﻳﺔ ﻟﻠﻨﺴﺒﺔ ﺑﻴﻦ ﻃﻮﻟﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻷﻗﺼﺮ ﻭﺍﻷﻃﻮﻝ ﻓﻲ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﺎﻧﻲ. ﻭﻫﻨﺎﻙ ﻋﺪﺓ ﺣﺎﻻﺕ ﻟﻠﺘﺸﺎﺑﻪ ﻣﻨﻬﺎ ﺯﺍﻭﻳﺘﻴﻦ ﻭﻳﺮﻣﺰ ﻟﻠﺘﺸﺎﺑﻪ ﺑﺎﻟﺮﻣﺰ (~)*
*ﻧﻈﺮﻳﺔ ﻓﻴﺜﺎﻏﻮﺭﺱ*
*ﻭﺍﺣﺪﺓ ﻣﻦ ﺍﻟﻨﻈﺮﻳﺎﺕ ﺍﻷﺳﺎﺳﻴﺔ ﻓﻲ ﺍﻟﻤﺜﻠﺜﺎﺕ ﻫﻲ **ﻧﻈﺮﻳﺔ ﻓﻴﺜﺎﻏﻮﺭﺱ*
* ﻭﺍﻟﺘﻲ ﺗﻨﺺ ﻋﻠﻰ ﺃﻧﻪ ﻓﻲ ﺍﻟﻤﺜﻠﺚ ﺍﻟﻘﺎﺋﻢ، ﻣﺮﺑﻊ ﻃﻮﻝ ﺍﻟﻮﺗﺮ (ﺍ َ) ﻳﺴﺎﻭﻱ ﻣﺠﻤﻮﻉ ﻣﺮﺑﻌﻲ ﻃﻮﻟﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻟﻘﺎﺋﻤﻴﻦ (ﺏ َ، ﺝ َ)، ﺃﻱ:*
*ﺃ َ2 = ﺏ َ2 + ﺝ َ2*
*"A"**2** = **"B"**2** + **"C"**2*
*ﻣﻤﺎ ﻳﻌﻨﻲ ﺃﻥ ﻣﻌﺮﻓﺔ ﻃﻮﻟﻲ ﺿﻠﻌﻴﻦ ﻣﻦ ﺍﻟﻤﺜﻠﺚ ﺍﻟﻘﺎﺋﻢ، ﻛﺎﻑٍ ﻟﻤﻌﺮﻓﺔ ﻃﻮﻝ ﺍﻟﻀﻠﻊ ﺍﻟﺜﺎﻟﺚ:*
*ﻣﻦ ﺍﻟﻤﻤﻜﻦ ﺗﻌﻤﻴﻢ ﻧﻈﺮﻳﺔ ﻓﻴﺜﺎﻏﻮﺭﺙ ﻟﺘﺸﻤﻞ ﺃﻱ ﻣﺜﻠﺚ ﻋﺒﺮ **ﻗﺎﻧﻮﻥ ﺟﻴﺐ ﺍﻟﺘﻤﺎﻡ*
*: ﺣﻴﺚ :*
*ﻣﺮﺑﻊ ﻃﻮﻝ ﺍﻟﻀﻠﻊ = ﻣﺠﻤﻮﻉ ﻣﺮﺑﻌﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻵﺧﺮﻳﻦ ﻣﻄﺮﻭﺡ ﻣﻨﻪ ﺿﻌﻒ ﺣﺎﺻﻞ ﺿﺮﻭﺏ ﻃﻮﻟﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻵﺧﺮﻳﻦ ﻓﻲ ﺟﻴﺐ ﺗﻤﺎﻡ "ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺤﺼﻮﺭﺓ ﺑﻴﻨﻬﻤﺎ"*
*|ﺃ|^2 = |ﺏ|^2 + |ﺝ|^2 - 2 × |ﺏ|× |ﺝ| × ﺟﺘﺎ (ﺩْ)*
*"A"**2** = **"B"**2** + **"C"**2** − 2 * **"B"** * **"C"** * **"cos"**α*
*ﻭ ﻫﻮ ﺻﺤﻴﺢ ﻟﻜﻞ ﺍﻟﻤﺜﻠﺜﺎﺕ ﺣﺘﻰ ﻭﻟﻮ ﻟﻢ ﺗﻜﻦ ﺍﻟﺰﺍﻭﻳﺔ (ﺩ) ﻗﺎﺋﻤﺔ.*
*ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ*
*ﺗﻌﻄﻰ **ﻣﺴﺎﺣﺔ*
* ﺍﻟﻤﺜﻠﺚ ﺑﺎﻟﻘﺎﻧﻮﻥ ﺍﻟﺘﺎﻟﻲ:*
*ﺍﻟﻤﺴﺎﺣﺔ = 0.5× ﻕ × ﻉ*
*"Area"** = 0.5 * **"B"** * **"H"*
*ﺣﻴﺚ (ﻕ ﺃﻭ **B**) ﻫﻲ ﻃﻮﻝ ﺃﺣﺪ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ( ﻭﻳﺴﻤﻰ ﺍﻟﻘﺎﻋﺪﺓ)، ﻭ(ﻉ ﺃﻭ **H**) ﻫﻮ ﻃﻮﻝ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻨﺎﺯﻝ ﻋﻠﻰ ﻫﺬﻩ ﺍﻟﻘﺎﻋﺪﺓ ﻣﻦ ﺍﻟﺮﺃﺱ ﺍﻟﻤﻘﺎﺑﻞ ﻟﻪ (ﻭﻳﺴﻤﻰ ﺍﻻﺭﺗﻔﺎﻉ).*
*ﻣﻦ ﺍﻟﻤﻤﻜﻦ ﺍﻟﺒﺮﻫﺎﻥ ﻋﻠﻰ ﺫﻟﻚ ﻣﻦ ﺧﻼﻝ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ:*
*ﻳﺤﻮﻝ ﺍﻟﻤﺜﻠﺚ ﺃﻭﻻ ﻟﻤﺘﻮﺍﺯﻱ ﺃﺿﻼﻉ**
**ﻣﺴﺎﺣﺘﻪ ﺿﻌﻒ ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ، ﺛﻢ ﺇﻟﻰ ﻣﺴﺘﻄﻴﻞ**.**ﻧﻘﺎﻁ ﻭﻣﺴﺘﻘﻴﻤﺎﺕ ﻭﺩﻭﺍﺋﺮ ﻣﺘﺼﻠﺔ ﺑﺎﻟﻤﺜﻠﺚ*
* *ﺍﻟﻤﻮﺳﻂ ﺍﻟﻌﻤﻮﺩﻱ*
* ﻟﻤﺜﻠﺚ ﻫﻮ ﻣﺴﺘﻘﻴﻢ ﻳﻤﺮ ﻣﻦ ﺃﺣﺪ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﻓﻲ ﻣﻨﺘﺼﻔﻪ ﻭﻳﻜﻮﻥ ﻋﻤﻮﺩﻳّﺎ ﻋﻠﻴﻪ ﻭﺗﺘﻼﻗﻰ ﺍﻟﻮﺳﻄﺎﺕ ﺍﻟﻌﻤﻮﺩﻳﺔ ﻟﻤﺜﻠﺚ ﻓﻲ ﻧﻘﻄﺔ ﺗﺴﻤﻰ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﺑﻤﺜﻠﺚ ﻭﻳﻜﻮﻥ ﻟﻬﺬﻩ ﺍﻟﻨﻘﻄﺔ ﻧﻔﺲ ﺍﻟﺒﻌﺪ ﻋﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﻼﺙ ﻭﻳﻜﻮﻥ ﺗﻘﺎﻃﻊ ﻣﻮﺳﻄﻴﻦ ﻋﻤﻮﺩﻳﻴﻦ ﻓﻘﻂ ﻛﺎﻓﻴﺎ ﻟﻤﻌﺮﻓﺔ ﻣﺮﻛﺰ ﻫﺬﻩ ﺍﻟﺪﺍﺋﺮﺓ.
**ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﻟﻤﺜﻠﺚ ﻳﻤﺮّ ﻣﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﻼﺙ*
* *ﺗﻘﻮﻝ **ﻣﺒﺮﻫﻨﺔ ﻃﺎﻟﺲ*
* ﺍﻧّﻪ ﺇﺫﺍ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﺑﺎﻟﻤﺜﻠﺚ ﺗﻮﺟﺪ ﻋﻠﻰ ﺿﻠﻊ ﻣﻦ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﻓﺎﻥّ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻘﺎﺑﻠﺔ ﻟﻬﺬﺍ ﺍﻟﻀﻠﻊ ﺗﻜﻮﻥ ﻗﺎﺋﻤﺔ.*
*ﻧﻘﻄﺔ ﺗﻘﺎﻃﻊ ﺍﻻﺭﺗﻔﺎﻋﺎﺕ ﻓﻲ ﻣﺜﻠﺚ ﺗﺴﻤﻰ **ﺍﻟﻤﺮﻛﺰ ﺍﻟﻘﺎﺋﻢ*
*.*
* *ﺍﻻﺭﺗﻔﺎﻉ*
* ﻫﻮ ﻣﺴﺘﻘﻴﻢ ﻳﻤﺮ ﺑﺮﺃﺱ ﻣﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﻭﺗﻜﻮﻥ ﻋﻤﻮﺩﻳﺔ ﻏﻠﻰ ﺍﻟﻀﻠﻊ ﺍﻟﻤﻘﺎﺑﻞ. ﻭﻳﻤﺜﻞ ﺍﻻﺭﺗﻔﺎﻉ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﺍﻟﺮﺃﺱ ﻭﺍﻟﻀﻠﻊ ﺍﻟﻤﻘﺎﺑﻞ ﻛﻤﺎ ﺗﺘﻘﺎﻃﻊ ﺍﻻﺭﺗﻔﺎﻋﺎﺕ ﻓﻲ ﻧﻘﻄﺔ ﺗﺴﻤﻰ **ﻣﺮﻛﺰ ﻗﺎﺋﻢ*
*.*
*ﺗﻘﺎﻃﻊ ﻣﻨﺼﻔﺎﺕ ﺍﻟﺰﻭﺍﻳﺎ ﻓﻲ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﺑﺎﻟﻤﺜﻠﺚ*
*.*
* *ﻣﻨﺼﻒ ﺍﻟﺰﺍﻭﻳﺔ ﻫﻮ ﻣﺴﺘﻘﻴﻢ ﻳﻤﺮّ ﻣﻦ ﺭﺃﺱ ﻣﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﻭﻳﻘﺴﻢ ﺍﻟﺰﺍﻭﻳﺔ ﺇﻟﻰ ﻧﺼﻔﻴﻦ ﻭﺗﺘﻘﺎﻃﻊ ﺍﻟﻤﻨﺼﻔﺎﺕ ﺍﻟﺜﻼﺛﺔ ﻓﻲ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﺎﻃﺔ ﺑﺎﻟﻤﺜﻠﺚ ﻭﻫﻲ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺘﻲ ﺗﻤﺲّ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﻼﺙ.*
* *ﺍﻟﻤﻮﺳّﻂ*
* ﻫﻮ **ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻢ*
* ﺗﻨﻄﻠﻖ ﻣﻦ ﺭﺃﺱ ﻣﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﻭﺗﻤﺮ ﻣﻦ ﻣﻨﺘﺼﻒ ﺍﻟﻀﻠﻊ ﺍﻟﻤﻘﺎﺑﻞ ﻭﺗﺘﻘﺎﻃﻊ ﺍﻟﻤﻮﺳﻄﺎﺕ ﺍﻟﺜﻼﺙ ﻓﻲ ﻧﻘﻄﺔ ﺗﺴﻤﻰ **ﻣﺮﻛﺰ ﺛﻘﻞ*
* ﺍﻟﻤﺜﻠﺚ ﻭﻳﻜﻮﻥ ﺗﻘﺎﻃﻊ ﻣﻮﺳﻄﻴﻦ ﻓﻘﻂ ﻛﺎﻓﻴﺎ ﻟﻤﻌﺮﻓﺔ ﻣﺮﻛﺰ ﺍﻟﺜﻘﻞ. ﻛﻤﺎ ﻳﻜﻮﻥ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﺭﺃﺱ ﺍﻟﻤﺜﻠﺚ **ﻭﻣﺮﻛﺰ ﺍﻟﺜﻘﻞ*
* ﻣﺴﺎﻭﻳﺎ ﻝ 2/3 ﺍﻟﻤﻮﺳﻂ ﺍﻟﺼﺎﺩﺭ ﻣﻦ ﺫﻟﻚ ﺍﻟﺮﺃﺱ.*
*ﺍﻟﻮﺳﻄﺎﺕ ﻭﻣﺮﻛﺰ ﺍﻟﺜﻘﻞ.*
* *ﻣﻨﺘﺼﻔﺎﺕ ﺍﻷﺿﻼﻉ ﺍﻟﺜﻼﺙ ﻭﻧﻘﻄﺔ ﺗﻘﺎﻃﻊ ﺍﻻﺭﺗﻔﺎﻉ ﻭﺍﻟﻀﻠﻊ ﺍﻟﻤﻘﺎﺑﻞ ﻟﻪ ﻣﻮﺟﻮﺩﺓ ﻛﻠﻬﺎ ﻋﻠﻰ ﻧﻔﺲ ﺍﻟﻤﺜﻠﺚ **ﺩﺍﺋﺮﺓ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﺴﻊ*
* ﻟﻠﻤﺜﻠﺚ ﻭﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺛﺔ ﺍﻟﻤﺘﺒﻘﻴﺔ ﻫﻲ ﻣﻨﺘﺼﻒ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﺭﺃﺱ ﺍﻟﻤﺜﻠﺚ **ﻭﺍﻟﻤﺮﻛﺰ ﺍﻟﻘﺎﺋﻢ*
*ﻭﺷﻌﺎﻉ*
* ﺩﺍﺋﺮﺓ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﺴﻊ ﻫﻲ ﻧﺼﻒ ﺷﻌﺎﻉ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﺑﺎﻟﻤﺜﻠﺚ.*
*ﺗﺴﻊ ﻧﻘﺎﻁ ﻣﻦ ﻫﺬﻩ ﺍﻟﺪﺍﺋﺮﺓ ﻣﻮﺟﻮﺩﺓ ﻋﻠﻰ ﺍﻟﻤﺜﻠﺚ.*
*ﺣﺴﺎﺏ ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ*
*ﺃﺑﺴﻂ ﻃﺮﻳﻘﺔ ﻟﺤﺴﺎﺏ ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ ﻭﺃﻛﺜﺮﻫﺎ ﺷﻬﺮﺓ ﻫﻲ*
*ﺣﻴﺚ **"S"** ﻫﻲ ﺍﻟﻤﺴﺎﺣﺔ ﻭ**"b"**ﻫﻲ ﻃﻮﻝ ﻗﺎﻋﺪﺓ ﺍﻟﻤﺜﻠﺚ ﻭ**"h"**ﻫﻮ ﺍﺭﺗﻔﺎﻉ ﺍﻟﻤﺜﻠﺚ. ﻗﺎﻋﺪﺓ ﺍﻟﻤﺜﻠﺚ ﺗﻤﺜﻞ ﺃﻱ ﺿﻠﻊ ﻣﻦ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﻭﺍﻻﺭﺗﻔﺎﻉ ﻫﻮ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺼﺎﺩﺭ ﻣﻦ ﺍﻟﺮﺃﺱ ﺍﻟﻤﻘﺎﺑﻞ ﻟﻠﻀﻠﻊ ﻭﺍﻟﻌﻤﻮﺩﻱّ ﻋﻠﻴﻪ.*
ﺍﻟﻤﻮﺿﻮﻉ ﺍﻻﺻﻠﻲ: ﺑﺤﺚ ﻟﻤﺎﺩﺓ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ(ﺍﻟﻤﺜﻠﺜﺎﺕ)
http://www.a1ash.com/vb/a1ash85056/#ixzz2G5Xc5pVp
*ﺗﺸﺎﺑﻪ ﻣﺜﻠﺜﻴﻦ*
*ﻳﻘﺎﻝ ﻋﻦ ﻣﺜﻠﺜﻴﻦ ﺃﻧﻬﻤﺎ ﻣﺘﺸﺎﺑﻬﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﺘﻘﺎﺑﻠﺔ ﻣﻦ ﻛﻞ ﻣﻨﻬﻤﺎ ﻣﺘﺴﺎﻭﻳﺔ، ﺃﻱ ﻋﻨﺪﻣﺎ ﻳﻨﺘﺞ ﺃﺣﺪﻫﻤﺎ ﻋﻦ ﺍﻵﺧﺮ ﺑﺘﻜﺒﻴﺮﻩ ﺃﻭ ﺗﺼﻐﻴﺮﻩ. ﻭﺗﻜﻮﻥ ﺃﻃﻮﺍﻝ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺜﻴﻦ ﺍﻟﻤﺘﺸﺎﺑﻬﻴﻦ ﻣﺘﻨﺎﺳﺒﺔ، ﺃﻱ ﺃﻧﻪ ﺇﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﺃﻗﺼﺮ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﺍﻷﻭﻝ ﻫﻮ ﺿﻌﻔﺎ ﻃﻮﻝ ﺃﻗﺼﺮ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﺎﻧﻲ، ﻓﺈﻥ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻷﻃﻮﻝ ﻭﺍﻟﻤﺘﻮﺳﻂ ﻣﻦ ﺍﻟﻤﺜﻠﺚ ﺍﻷﻭﻝ ﻫﻮ ﺿﻌﻔﺎ ﻃﻮﻟﻲ ﻟﻀﻠﻌﻴﻦ ﺍﻷﻃﻮﻝ ﻭﺍﻟﻤﺘﻮﺳﻂ ﻣﻦ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﺎﻧﻲ ﺃﻳﻀﺎ، ﻭﺑﺎﻟﺘﺎﻟﻲ ﻓﺎﻥ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻃﻮﻟﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻷﻗﺼﺮ ﻭﺍﻷﻃﻮﻝ ﻓﻲ ﺍﻟﻤﺜﻠﺚ ﺍﻷﻭﻝ ﻣﺴﺎﻭﻳﺔ ﻟﻠﻨﺴﺒﺔ ﺑﻴﻦ ﻃﻮﻟﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻷﻗﺼﺮ ﻭﺍﻷﻃﻮﻝ ﻓﻲ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﺎﻧﻲ. ﻭﻫﻨﺎﻙ ﻋﺪﺓ ﺣﺎﻻﺕ ﻟﻠﺘﺸﺎﺑﻪ ﻣﻨﻬﺎ ﺯﺍﻭﻳﺘﻴﻦ ﻭﻳﺮﻣﺰ ﻟﻠﺘﺸﺎﺑﻪ ﺑﺎﻟﺮﻣﺰ (~)*
*ﻧﻈﺮﻳﺔ ﻓﻴﺜﺎﻏﻮﺭﺱ*
*ﻭﺍﺣﺪﺓ ﻣﻦ ﺍﻟﻨﻈﺮﻳﺎﺕ ﺍﻷﺳﺎﺳﻴﺔ ﻓﻲ ﺍﻟﻤﺜﻠﺜﺎﺕ ﻫﻲ **ﻧﻈﺮﻳﺔ ﻓﻴﺜﺎﻏﻮﺭﺱ*
* ﻭﺍﻟﺘﻲ ﺗﻨﺺ ﻋﻠﻰ ﺃﻧﻪ ﻓﻲ ﺍﻟﻤﺜﻠﺚ ﺍﻟﻘﺎﺋﻢ، ﻣﺮﺑﻊ ﻃﻮﻝ ﺍﻟﻮﺗﺮ (ﺍ َ) ﻳﺴﺎﻭﻱ ﻣﺠﻤﻮﻉ ﻣﺮﺑﻌﻲ ﻃﻮﻟﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻟﻘﺎﺋﻤﻴﻦ (ﺏ َ، ﺝ َ)، ﺃﻱ:*
*ﺃ َ2 = ﺏ َ2 + ﺝ َ2*
*"A"**2** = **"B"**2** + **"C"**2*
*ﻣﻤﺎ ﻳﻌﻨﻲ ﺃﻥ ﻣﻌﺮﻓﺔ ﻃﻮﻟﻲ ﺿﻠﻌﻴﻦ ﻣﻦ ﺍﻟﻤﺜﻠﺚ ﺍﻟﻘﺎﺋﻢ، ﻛﺎﻑٍ ﻟﻤﻌﺮﻓﺔ ﻃﻮﻝ ﺍﻟﻀﻠﻊ ﺍﻟﺜﺎﻟﺚ:*
*ﻣﻦ ﺍﻟﻤﻤﻜﻦ ﺗﻌﻤﻴﻢ ﻧﻈﺮﻳﺔ ﻓﻴﺜﺎﻏﻮﺭﺙ ﻟﺘﺸﻤﻞ ﺃﻱ ﻣﺜﻠﺚ ﻋﺒﺮ **ﻗﺎﻧﻮﻥ ﺟﻴﺐ ﺍﻟﺘﻤﺎﻡ*
*: ﺣﻴﺚ :*
*ﻣﺮﺑﻊ ﻃﻮﻝ ﺍﻟﻀﻠﻊ = ﻣﺠﻤﻮﻉ ﻣﺮﺑﻌﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻵﺧﺮﻳﻦ ﻣﻄﺮﻭﺡ ﻣﻨﻪ ﺿﻌﻒ ﺣﺎﺻﻞ ﺿﺮﻭﺏ ﻃﻮﻟﻲ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻵﺧﺮﻳﻦ ﻓﻲ ﺟﻴﺐ ﺗﻤﺎﻡ "ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺤﺼﻮﺭﺓ ﺑﻴﻨﻬﻤﺎ"*
*|ﺃ|^2 = |ﺏ|^2 + |ﺝ|^2 - 2 × |ﺏ|× |ﺝ| × ﺟﺘﺎ (ﺩْ)*
*"A"**2** = **"B"**2** + **"C"**2** − 2 * **"B"** * **"C"** * **"cos"**α*
*ﻭ ﻫﻮ ﺻﺤﻴﺢ ﻟﻜﻞ ﺍﻟﻤﺜﻠﺜﺎﺕ ﺣﺘﻰ ﻭﻟﻮ ﻟﻢ ﺗﻜﻦ ﺍﻟﺰﺍﻭﻳﺔ (ﺩ) ﻗﺎﺋﻤﺔ.*
*ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ*
*ﺗﻌﻄﻰ **ﻣﺴﺎﺣﺔ*
* ﺍﻟﻤﺜﻠﺚ ﺑﺎﻟﻘﺎﻧﻮﻥ ﺍﻟﺘﺎﻟﻲ:*
*ﺍﻟﻤﺴﺎﺣﺔ = 0.5× ﻕ × ﻉ*
*"Area"** = 0.5 * **"B"** * **"H"*
*ﺣﻴﺚ (ﻕ ﺃﻭ **B**) ﻫﻲ ﻃﻮﻝ ﺃﺣﺪ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ( ﻭﻳﺴﻤﻰ ﺍﻟﻘﺎﻋﺪﺓ)، ﻭ(ﻉ ﺃﻭ **H**) ﻫﻮ ﻃﻮﻝ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻨﺎﺯﻝ ﻋﻠﻰ ﻫﺬﻩ ﺍﻟﻘﺎﻋﺪﺓ ﻣﻦ ﺍﻟﺮﺃﺱ ﺍﻟﻤﻘﺎﺑﻞ ﻟﻪ (ﻭﻳﺴﻤﻰ ﺍﻻﺭﺗﻔﺎﻉ).*
*ﻣﻦ ﺍﻟﻤﻤﻜﻦ ﺍﻟﺒﺮﻫﺎﻥ ﻋﻠﻰ ﺫﻟﻚ ﻣﻦ ﺧﻼﻝ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ:*
*ﻳﺤﻮﻝ ﺍﻟﻤﺜﻠﺚ ﺃﻭﻻ ﻟﻤﺘﻮﺍﺯﻱ ﺃﺿﻼﻉ**
**ﻣﺴﺎﺣﺘﻪ ﺿﻌﻒ ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ، ﺛﻢ ﺇﻟﻰ ﻣﺴﺘﻄﻴﻞ**.**ﻧﻘﺎﻁ ﻭﻣﺴﺘﻘﻴﻤﺎﺕ ﻭﺩﻭﺍﺋﺮ ﻣﺘﺼﻠﺔ ﺑﺎﻟﻤﺜﻠﺚ*
* *ﺍﻟﻤﻮﺳﻂ ﺍﻟﻌﻤﻮﺩﻱ*
* ﻟﻤﺜﻠﺚ ﻫﻮ ﻣﺴﺘﻘﻴﻢ ﻳﻤﺮ ﻣﻦ ﺃﺣﺪ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﻓﻲ ﻣﻨﺘﺼﻔﻪ ﻭﻳﻜﻮﻥ ﻋﻤﻮﺩﻳّﺎ ﻋﻠﻴﻪ ﻭﺗﺘﻼﻗﻰ ﺍﻟﻮﺳﻄﺎﺕ ﺍﻟﻌﻤﻮﺩﻳﺔ ﻟﻤﺜﻠﺚ ﻓﻲ ﻧﻘﻄﺔ ﺗﺴﻤﻰ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﺑﻤﺜﻠﺚ ﻭﻳﻜﻮﻥ ﻟﻬﺬﻩ ﺍﻟﻨﻘﻄﺔ ﻧﻔﺲ ﺍﻟﺒﻌﺪ ﻋﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﻼﺙ ﻭﻳﻜﻮﻥ ﺗﻘﺎﻃﻊ ﻣﻮﺳﻄﻴﻦ ﻋﻤﻮﺩﻳﻴﻦ ﻓﻘﻂ ﻛﺎﻓﻴﺎ ﻟﻤﻌﺮﻓﺔ ﻣﺮﻛﺰ ﻫﺬﻩ ﺍﻟﺪﺍﺋﺮﺓ.
**ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﻟﻤﺜﻠﺚ ﻳﻤﺮّ ﻣﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﻼﺙ*
* *ﺗﻘﻮﻝ **ﻣﺒﺮﻫﻨﺔ ﻃﺎﻟﺲ*
* ﺍﻧّﻪ ﺇﺫﺍ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﺑﺎﻟﻤﺜﻠﺚ ﺗﻮﺟﺪ ﻋﻠﻰ ﺿﻠﻊ ﻣﻦ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﻓﺎﻥّ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻘﺎﺑﻠﺔ ﻟﻬﺬﺍ ﺍﻟﻀﻠﻊ ﺗﻜﻮﻥ ﻗﺎﺋﻤﺔ.*
*ﻧﻘﻄﺔ ﺗﻘﺎﻃﻊ ﺍﻻﺭﺗﻔﺎﻋﺎﺕ ﻓﻲ ﻣﺜﻠﺚ ﺗﺴﻤﻰ **ﺍﻟﻤﺮﻛﺰ ﺍﻟﻘﺎﺋﻢ*
*.*
* *ﺍﻻﺭﺗﻔﺎﻉ*
* ﻫﻮ ﻣﺴﺘﻘﻴﻢ ﻳﻤﺮ ﺑﺮﺃﺱ ﻣﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﻭﺗﻜﻮﻥ ﻋﻤﻮﺩﻳﺔ ﻏﻠﻰ ﺍﻟﻀﻠﻊ ﺍﻟﻤﻘﺎﺑﻞ. ﻭﻳﻤﺜﻞ ﺍﻻﺭﺗﻔﺎﻉ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﺍﻟﺮﺃﺱ ﻭﺍﻟﻀﻠﻊ ﺍﻟﻤﻘﺎﺑﻞ ﻛﻤﺎ ﺗﺘﻘﺎﻃﻊ ﺍﻻﺭﺗﻔﺎﻋﺎﺕ ﻓﻲ ﻧﻘﻄﺔ ﺗﺴﻤﻰ **ﻣﺮﻛﺰ ﻗﺎﺋﻢ*
*.*
*ﺗﻘﺎﻃﻊ ﻣﻨﺼﻔﺎﺕ ﺍﻟﺰﻭﺍﻳﺎ ﻓﻲ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﺑﺎﻟﻤﺜﻠﺚ*
*.*
* *ﻣﻨﺼﻒ ﺍﻟﺰﺍﻭﻳﺔ ﻫﻮ ﻣﺴﺘﻘﻴﻢ ﻳﻤﺮّ ﻣﻦ ﺭﺃﺱ ﻣﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﻭﻳﻘﺴﻢ ﺍﻟﺰﺍﻭﻳﺔ ﺇﻟﻰ ﻧﺼﻔﻴﻦ ﻭﺗﺘﻘﺎﻃﻊ ﺍﻟﻤﻨﺼﻔﺎﺕ ﺍﻟﺜﻼﺛﺔ ﻓﻲ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﺎﻃﺔ ﺑﺎﻟﻤﺜﻠﺚ ﻭﻫﻲ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺘﻲ ﺗﻤﺲّ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﻼﺙ.*
* *ﺍﻟﻤﻮﺳّﻂ*
* ﻫﻮ **ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻢ*
* ﺗﻨﻄﻠﻖ ﻣﻦ ﺭﺃﺱ ﻣﻦ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﻭﺗﻤﺮ ﻣﻦ ﻣﻨﺘﺼﻒ ﺍﻟﻀﻠﻊ ﺍﻟﻤﻘﺎﺑﻞ ﻭﺗﺘﻘﺎﻃﻊ ﺍﻟﻤﻮﺳﻄﺎﺕ ﺍﻟﺜﻼﺙ ﻓﻲ ﻧﻘﻄﺔ ﺗﺴﻤﻰ **ﻣﺮﻛﺰ ﺛﻘﻞ*
* ﺍﻟﻤﺜﻠﺚ ﻭﻳﻜﻮﻥ ﺗﻘﺎﻃﻊ ﻣﻮﺳﻄﻴﻦ ﻓﻘﻂ ﻛﺎﻓﻴﺎ ﻟﻤﻌﺮﻓﺔ ﻣﺮﻛﺰ ﺍﻟﺜﻘﻞ. ﻛﻤﺎ ﻳﻜﻮﻥ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﺭﺃﺱ ﺍﻟﻤﺜﻠﺚ **ﻭﻣﺮﻛﺰ ﺍﻟﺜﻘﻞ*
* ﻣﺴﺎﻭﻳﺎ ﻝ 2/3 ﺍﻟﻤﻮﺳﻂ ﺍﻟﺼﺎﺩﺭ ﻣﻦ ﺫﻟﻚ ﺍﻟﺮﺃﺱ.*
*ﺍﻟﻮﺳﻄﺎﺕ ﻭﻣﺮﻛﺰ ﺍﻟﺜﻘﻞ.*
* *ﻣﻨﺘﺼﻔﺎﺕ ﺍﻷﺿﻼﻉ ﺍﻟﺜﻼﺙ ﻭﻧﻘﻄﺔ ﺗﻘﺎﻃﻊ ﺍﻻﺭﺗﻔﺎﻉ ﻭﺍﻟﻀﻠﻊ ﺍﻟﻤﻘﺎﺑﻞ ﻟﻪ ﻣﻮﺟﻮﺩﺓ ﻛﻠﻬﺎ ﻋﻠﻰ ﻧﻔﺲ ﺍﻟﻤﺜﻠﺚ **ﺩﺍﺋﺮﺓ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﺴﻊ*
* ﻟﻠﻤﺜﻠﺚ ﻭﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺛﺔ ﺍﻟﻤﺘﺒﻘﻴﺔ ﻫﻲ ﻣﻨﺘﺼﻒ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﺭﺃﺱ ﺍﻟﻤﺜﻠﺚ **ﻭﺍﻟﻤﺮﻛﺰ ﺍﻟﻘﺎﺋﻢ*
*ﻭﺷﻌﺎﻉ*
* ﺩﺍﺋﺮﺓ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﺴﻊ ﻫﻲ ﻧﺼﻒ ﺷﻌﺎﻉ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﺑﺎﻟﻤﺜﻠﺚ.*
*ﺗﺴﻊ ﻧﻘﺎﻁ ﻣﻦ ﻫﺬﻩ ﺍﻟﺪﺍﺋﺮﺓ ﻣﻮﺟﻮﺩﺓ ﻋﻠﻰ ﺍﻟﻤﺜﻠﺚ.*
*ﺣﺴﺎﺏ ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ*
*ﺃﺑﺴﻂ ﻃﺮﻳﻘﺔ ﻟﺤﺴﺎﺏ ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ ﻭﺃﻛﺜﺮﻫﺎ ﺷﻬﺮﺓ ﻫﻲ*
*ﺣﻴﺚ **"S"** ﻫﻲ ﺍﻟﻤﺴﺎﺣﺔ ﻭ**"b"**ﻫﻲ ﻃﻮﻝ ﻗﺎﻋﺪﺓ ﺍﻟﻤﺜﻠﺚ ﻭ**"h"**ﻫﻮ ﺍﺭﺗﻔﺎﻉ ﺍﻟﻤﺜﻠﺚ. ﻗﺎﻋﺪﺓ ﺍﻟﻤﺜﻠﺚ ﺗﻤﺜﻞ ﺃﻱ ﺿﻠﻊ ﻣﻦ ﺃﺿﻼﻉ ﺍﻟﻤﺜﻠﺚ ﻭﺍﻻﺭﺗﻔﺎﻉ ﻫﻮ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺼﺎﺩﺭ ﻣﻦ ﺍﻟﺮﺃﺱ ﺍﻟﻤﻘﺎﺑﻞ ﻟﻠﻀﻠﻊ ﻭﺍﻟﻌﻤﻮﺩﻱّ ﻋﻠﻴﻪ.*
ﺍﻟﻤﻮﺿﻮﻉ ﺍﻻﺻﻠﻲ: ﺑﺤﺚ ﻟﻤﺎﺩﺓ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ(ﺍﻟﻤﺜﻠﺜﺎﺕ)
http://www.a1ash.com/vb/a1ash85056/#ixzz2G5Xc5pVp
