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In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier in 1782, and got the name orthocentric tetrahedron by G. de Longchamps in 1890.〔.〕 In an orthocentric tetrahedron the four altitudes are concurrent. This common point is called the orthocenter, and it has the property that it is the symmetric point of the center of the circumscribed sphere with respect to the centroid.〔 Hence the orthocenter coincides with the Monge point of the tetrahedron. ==Characterizations== All tetrahedra can be inscribed in a parallelepiped. A tetrahedron is orthocentric if and only if its circumscribed parallelepiped is a rhombohedron. Indeed, in any tetrahedron, a pair of opposite edges is perpendicular if and only if the corresponding faces of the circumscribed parallelepiped are rhombi. If four faces of a parallelepiped are rhombi, then all edges have equal lengths and all six faces are rhombi; it follows that if two pairs of opposite edges in a tetrahedron are perpendicular, then so is the third pair, and the tetrahedron is orthocentric.〔 A tetrahedron ''ABCD'' is orthocentric if and only if the sum of the squares of opposite edges is the same for the three pairs of opposite edges:〔Reiman, István, "International Mathematical Olympiad: 1976-1990", Anthem Press, 2005, pp. 175-176.〕〔 : In fact, it is enough for only two pairs of opposite edges to satisfy this condition for the tetrahedron to be orthocentric. Another necessary and sufficient condition for a tetrahedron to be orthocentric is that its three bimedians have equal length.〔Hazewinkel, Michiel, "Encyclopaedia of mathematics: Supplement, Volym 3", Kluwer Academic Publishers, 1997, p. 468.〕 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 tetrahedron」の詳細全文を読む スポンサード リンク
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