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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 Droz-Farny line theorem ： ウィキペディア英語版 Droz-Farny line theorem In Euclidean geometry, the Droz-Farny line theorem is a property of two perpendicular lines through the orthocenter of an arbitrary triangle. Let $T$ be a triangle with vertices $A$, $B$, and $C$, and let $H$ be its orthocenter (the common point of its three altitude lines. Let $L\_1$ and $L\_2$ be any two mutually perpendicular lines through $H$. Let $A\_1$, $B\_1$, and $C\_1$ be the points where $L\_1$ intersects the side lines $BC$, $CA$, and $AB$, respectively. Similarly, let Let $A\_2$, $B\_2$, and $C\_2$ be the points where $L\_2$ intersects those side lines. The Droz-Farny line theorem says that the midpoints three segments $A\_1A\_2$, $B\_1B\_2$, and $C\_1C\_2$ are collinear.〔〔〔 The theorem was stated by Arnold Droz-Farny in 1899,〔 but it is not clear whether he had a proof.〔 ==Goormaghtigh's generalization== A generalization of the Droz-Farny line theorem was proved in 1930 by René Goormaghtigh.〔 As above, let $T$ be a triangle with vertices $A$, $B$, and $C$. Let $P$ be any point distinct from $A$, $B$, and $C$, and $L$ be any line through $P$. Let $A\_1$, $B\_1$, and $C\_1$ be points on the side lines $BC$, $CA$, and $AB$, respectively, such that the lines $PA\_1$, $PB\_1$, and $PC\_1$ are the images of the lines $PA$, $PB$, and $PC$, respectively, by reflection against the line $L$. Goormaghtigh's theorem then says that the points $A\_1$, $B\_1$, and $C\_1$ are collinear. The Droz-Farny line theorem is a special case of this result, when $P$ is the orthocenter of triangle $T$. 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「Droz-Farny line 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.