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In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points and lie on the circle at infinity. These vertices can be called ideal vertices. In the hyperbolic metric, any two ideal triangles are congruent. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. == Properties == In the standard hyperbolic plane (with Gaussian curvature -1 at every point): * The interior angles of an ideal triangle are all zero. * Any ideal triangle has area π. * Any ideal triangle has infinite perimeter. * The inscribed circle to an ideal triangle meets the triangle in three points of tangency, forming an equilateral triangle with side length :: :where is the golden ratio.〔(【引用サイトリンク】 title=Modèle hyperbolique de Klein - Beltrami )〕 * The distance from any point in the triangle to the second-closest side of the triangle is less than or equal to ''d'', with equality only for the three equilateral triangle vertices described above. The same inequality holds for hyperbolic triangles more generally; in a non-ideal triangle, the distance to the second-closest side is strictly less than ''d''. If the curvature is −''K'' everywhere rather than −1, the areas above should be multiplied by 1/''K'' and the lengths and distances should be multiplied by 1/√''K''. 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「Ideal triangle」の詳細全文を読む スポンサード リンク
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