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In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. If four points form an orthocentric system, then ''each'' of the four points is the orthocenter of the other three. These four possible triangles will all have the same nine-point circle. Consequently these four possible triangles must all have circumcircles with the same circumradius. ==The common nine-point circle== The center of this common nine-point circle lies at the centroid of the four orthocentric points. The radius of the common nine-point circle is the distance from the nine-point center to the midpoint of any of the six connectors that join any pair of orthocentric points through which the common nine-point circle passes. The nine-point circle also passes through the three orthogonal intersections at the feet of the altitudes of the four possible triangle. This common nine-point center lies at the midpoint of the connector that joins any orthocentric point to the circumcenter of the triangle formed from the other three orthocentric points. The common nine-point circle is tangent to all 16 incircles and excircles of the four triangles whose vertices form the orthocentric system.〔Weisstein, Eric W. "Orthocentric System." From MathWorld--A Wolfram Web Resource. ()〕 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「Orthocentric system」の詳細全文を読む スポンサード リンク
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