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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 Isodynamic point ： ウィキペディア英語版 Isodynamic point In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the distances from the isodynamic point to the triangle vertices are inversely proportional to the opposite side lengths of the triangle. Triangles that are similar to each other have isodynamic points in corresponding locations in the plane, so the isodynamic points are triangle centers, and unlike other triangle centers the isodynamic points are also invariant under Möbius transformations. A triangle that is itself equilateral has a unique isodynamic point, at its centroid; every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by .〔For the credit to Neuberg, see e.g. and .〕 ==Distance ratios== The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points. If $S$ and $S\text{'}$ are the isodynamic points of a triangle $ABC$, then the three products of distances $AS\backslash cdot\; BC=BS\backslash cdot\; AC=CS\backslash cdot\; AB$ are equal. The analogous equalities also hold for $S\text{'}$.〔 states that this property is the reason for calling these points "isodynamic".〕 Equivalently to the product formula, the distances $AS$, $BS$, and $CS$ are inversely proportional to the corresponding triangle side lengths $BC$, $AC$, and $AB$. $S$ and $S\text{'}$ are the common intersection points of the three circles of Apollonius associated with triangle of a triangle $ABC$, the three circles that each pass through one vertex of the triangle and maintain a constant ratio of distances to the other two vertices.〔 Hence, line $SS\text{'}$ is the common radical axis for each of the three pairs of circles of Apollonius. The perpendicular bisector of line segment $SS\text{'}$ is the Lemoine line, which contains the three centers of the circles of Apollonius.〔 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「Isodynamic point」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.