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In geometry, the isoperimetric point is a special point associated with a plane triangle. The term was originally introduced by G.R. Veldkamp in a paper published in the American Mathematical Monthly in 1985 to denote a point ''P'' in the plane of a triangle ''ABC'' having the property that the triangles ''PBC'', ''PCA'' and ''PAB'' have isoperimeters, that is, having the property that :''PB'' + ''BC'' + ''CP'' = ''PC'' + ''CA'' + ''AP'' = ''PA'' + ''AB'' + ''BP''. Isoperimetric points in the sense of Veldkamp exist only for triangles satisfying certain conditions. The isoperimetric point of triangle ''ABC'' in the sense of Veldkamp, if it exists, has the following trilinear coordinates. : ( sec ( ''A''/2 ) cos ( ''B''/2 ) cos ( ''C''/2 ) − 1 , sec ( ''B''/2 ) cos ( ''C''/2 ) cos ( ''A''/2 ) − 1 , sec ( ''C''/2 ) cos ( ''A''/2 ) cos ( ''B''/2 ) − 1 ) Given any triangle ''ABC'' one can associate with it a point ''P'' having trilinear coordinates as given above. This point is a triangle center and in Clark Kimberling's Encyclopedia of Triangle Centers (ETC) it is called the isoperimetric point of the triangle ''ABC''. It is designated as the triangle center X(175).〔 The point X(175) need not be an isoperimetric point of triangle ''ABC'' in the sense of Veldkamp. However, if isoperimetric point of triangle ''ABC'' in the sense of Veldkamp exists, then it would be identical to the point X(175). The point ''P'' with the property that the triangles ''PBC'', ''PCA'' and ''PAB'' have equal perimeters has been studied as early as 1890 in an article by Emile Lemoine.〔The article by Emile Lemoine can be accessed in Gallica. The paper begins at page 111 and the point is discussed in page 126.(Gallica )〕 ==Existence of isoperimetric point in the sense of Veldkamp== Let ''ABC'' be any triangle. Let the sidelengths of this triangle be ''a'', ''b'', and ''c''. Let its circumradius be ''R'' and inradius be ''r''. The necessary and sufficient condition for the existence of an isoperimetric point in the sense of Veldkamp can be stated as follows.〔 :The triangle ''ABC'' has an isoperimetric point in the sense of Veldkamp if and only if ''a'' + ''b'' + ''c'' > 4''R'' + ''r''. For all acute angled triangles ''ABC'' we have ''a'' + ''b'' + ''c'' > 2''R'' + ''r'' > 4''R'' + ''r'', and so all acute angled triangles have isoperimetric points in the sense of Veldkamp. 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「Isoperimetric point」の詳細全文を読む スポンサード リンク
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