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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 Butterfly theorem ： ウィキペディア英語版 Butterfly theorem The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:〔Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929).〕 Let ''M'' be the midpoint of a chord ''PQ'' of a circle, through which two other chords ''AB'' and ''CD'' are drawn; ''AD'' and ''BC'' intersect chord ''PQ'' at ''X'' and ''Y'' correspondingly. Then ''M'' is the midpoint of ''XY''. ==Proof== A formal proof of the theorem is as follows: Let the perpendiculars $XX\text{'}\backslash ,$ and $XX\text{'}\text{'}\backslash ,$ be dropped from the point $X\backslash ,$ on the straight lines $AM\backslash ,$ and $DM\backslash ,$ respectively. Similarly, let $YY\text{'}\backslash ,$ and $YY\text{'}\text{'}\backslash ,$ be dropped from the point $Y\backslash ,$ perpendicular to the straight lines $BM\backslash ,$ and $CM\backslash ,$ respectively. Now, since :: $\backslash triangle\; MXX\text{'}\; \backslash sim\; \backslash triangle\; MYY\text{'},\backslash ,$ : $=\; ,$ :: $\backslash triangle\; MXX\text{'}\text{'}\; \backslash sim\; \backslash triangle\; MYY\text{'}\text{'},\backslash ,$ : $=\; ,$ :: $\backslash triangle\; AXX\text{'}\; \backslash sim\; \backslash triangle\; CYY\text{'}\text{'},\backslash ,$ : $=\; ,$ :: $\backslash triangle\; DXX\text{'}\text{'}\; \backslash sim\; \backslash triangle\; BYY\text{'},\backslash ,$ : $=\; ,$ From the preceding equations, it can be easily seen that : $\backslash left(\backslash right)^2\; =\; ,$ : $=\; ,$ : $=\; ,$ since $PM\; \backslash ,$ = $MQ\; \backslash ,$ Now, :$=\; .$ So, it can be concluded that $MX\; =\; MY,\; \backslash ,$ or $M\; \backslash ,$ is the midpoint of $XY.\; \backslash ,$ An alternate proof using projective geometry can be found in problem 8 of the link below. http://www.imomath.com/index.php?options=628&lmm=0 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「Butterfly 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.