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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 Brocard points ： ウィキペディア英語版 Brocard points In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845 – 1922), a French mathematician. ==Definition== In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled ''A'', ''B'' and ''C'' in anticlockwise order, there is exactly one point ''P'' such that the line segments ''AP'', ''BP'', and ''CP'' form the same angle, ω, with the respective sides ''c'', ''a'', and ''b'', namely that : $\backslash angle\; PAB\; =\; \backslash angle\; PBC\; =\; \backslash angle\; PCA.\backslash ,$ Point ''P'' is called the first Brocard point of the triangle ''ABC'', and the angle ''ω'' is called the Brocard angle of the triangle. The following applies to this angle: :$\backslash cot\backslash omega\; =\; \backslash cot\; \backslash alpha\; +\; \backslash cot\; \backslash beta\; +\; \backslash cot\; \backslash gamma.\backslash ,$ There is also a second Brocard point, Q, in triangle ''ABC'' such that line segments ''AQ'', ''BQ'', and ''CQ'' form equal angles with sides ''b'', ''c'', and ''a'' respectively. In other words, the equations $\backslash angle\; QCB\; =\; \backslash angle\; QBA\; =\; \backslash angle\; QAC$ apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words angle $\backslash angle\; PBC\; =\; \backslash angle\; PCA\; =\; \backslash angle\; PAB$ is the same as $\backslash angle\; QCB\; =\; \backslash angle\; QBA\; =\; \backslash angle\; QAC.$ The two Brocard points are closely related to one another; In fact, the difference between the first and the second depends on the order in which the angles of triangle ''ABC'' are taken. So for example, the first Brocard point of triangle ''ABC'' is the same as the second Brocard point of triangle ''ACB''. The two Brocard points of a triangle ''ABC'' are isogonal conjugates of each other. 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「Brocard points」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.