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In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the three sides of the triangle (these may need to be produced, i.e., extended). Label ''L'', ''M'', ''N'' the intersections of the lines from ''P'' with the sides ''BC'', ''AC'', ''AB''. The pedal triangle is then ''LMN''. The location of the chosen point ''P'' relative to the chosen triangle ''ABC'' gives rise to some special cases: * If ''P = ''orthocenter, then ''LMN = ''orthic triangle. * If ''P = ''incenter, then ''LMN = ''intouch triangle. If ''P'' is on the circumcircle of the triangle, ''LMN'' collapses to a line. This is then called the pedal line, or sometimes the Simson line after Robert Simson. The vertices of the pedal triangle of an interior point ''P'', as shown in the top diagram, divide the sides of the original triangle in such a way as to satisfy〔Alfred S. Posamentier and Charles T. Salkind, ''Challenging Problems in Geometry'', Dover Publishing Co., second revised edition, 1996.〕 : ==Trilinear coordinates== If ''P'' has trilinear coordinates ''p'' : ''q'' : ''r'', then the vertices ''L,M,N'' of the pedal triangle of ''P'' are given by *''L = 0 : q + p'' cos C'' : r + p ''cos'' B'' *''M = p + q ''cos'' C : 0 : r + q ''cos'' A'' *''N = p + r ''cos'' B : q + r ''cos'' A : 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「Pedal triangle」の詳細全文を読む スポンサード リンク
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