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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 Routh's theorem ： ウィキペディア英語版 Routh's theorem In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle $ABC$ points $D$, $E$, and $F$ lie on segments $BC$, $CA$, and $AB$, then writing $\backslash tfrac$$=\; x$, $\backslash tfrac$$=\; y$, and $\backslash tfrac$$=\; z$, the signed area of the triangle formed by the cevians $AD$, $BE$, and $CF$ is the area of triangle $ABC$ times : $\backslash frac.$ This theorem was given by Edward John Routh on page 82 of his ''Treatise on Analytical Statics with Numerous Examples'' in 1896. The particular case $x\; =\; y\; =\; z\; =\; 2$ has become popularized as the one-seventh area triangle. The $x\; =\; y\; =\; z\; =\; 1$ case implies that the three medians are concurrent (through the centroid). ==Proof== Suppose the area of triangle$ABC$ is 1. For triangle$ABD$ and line$FRC$ using Menelaus's theorem, We could obtain: :$\backslash frac\; \backslash times\; \backslash frac\; \backslash times\; \backslash frac\; =\; 1$ Then$\backslash frac\; =\; \backslash frac\; \backslash times\; \backslash frac\; =\; \backslash frac$ So the area of triangle$ARC$ is: :$S\_\; =\; \backslash frac\; S\_\; =\; \backslash frac\; \backslash times\; \backslash frac\; S\_\; =\; \backslash frac$ Similarly, we could know: $S\_\; =\; \backslash frac$ and $S\_\; =\; \backslash frac$ Thus the area of triangle$PQR$ is: :$\backslash displaystyle\; S\_\; =\; S\_\; -\; S\_\; -\; S\_\; -\; S\_$ :$=\; 1\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac$ : $=\backslash frac.$ 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「Routh'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.