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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pompeiu's theorem ： ウィキペディア英語版 Pompeiu's theorem Pompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is quite simple, but not classical. It states the following: :''Given an equilateral triangle ABC in the plane, and a point P in the plane of the triangle ABC, the lengths PA, PB, and PC form the sides of a triangle(If and only if P does not lie on circumcircle of the triangle).'' If Point P is on the circumcircle then PA PB PC will form a triangle with area 0 . That is sum of 2 Sides will be equal to the 3rd side ! The proof is quick. Consider a rotation of 60° about the point ''C''. Assume ''A'' maps to ''B'', and ''P'' maps to ''P'' '. Then we have $\backslash scriptstyle\; PC\backslash \; =\backslash \; P\text{'}C$, and $\backslash scriptstyle\backslash angle\; PCP\text{'}\backslash \; =\backslash \; 60^$. Hence triangle ''PCP'' ' is equilateral and $\backslash scriptstyle\; PP\text{'}\backslash \; =\backslash \; PC$. It is obvious that $\backslash scriptstyle\; PA\backslash \; =\backslash \; P\text{'}B$. Thus, triangle ''PBP'' ' has sides equal to ''PA'', ''PB'', and ''PC'' and the proof by construction is complete. Further investigations reveal that if ''P'' is not in the interior of the triangle, but rather on the circumcircle, then ''PA'', ''PB'', ''PC'' form a degenerate triangle, with the largest being equal to the sum of the others. . *To study it deep go for MOC by Titu Andrescu . ==External links== *(MathWorld's page on Pompeiu's Theorem ) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pompeiu's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.