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In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron. ==Triangle centers== Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear.〔 Reprinted in ''Opera Omnia'', ser. I, vol. XXVI, pp. 139–157, Societas Scientiarum Naturalium Helveticae, Lausanne, 1953, . Summarized at: (Dartmouth College. ) 〕 This property is also true for another triangle center, the nine-point center, although it had not been defined in Euler's time. In equilateral triangles, these four points coincide, but in any other triangle they are all distinct from each other, and the Euler line is determined by any two of them. Other notable points that lie on the Euler line include the de Longchamps point, the Schiffler point, and the Exeter point.〔 However, the incenter generally does not lie on the Euler line; it is on the Euler line only for isosceles triangles,〔.〕 for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers. The tangential triangle of a reference triangle is tangent to the latter's circumcircle at the reference triangle's vertices. The circumcenter of the tangential triangle lies on the Euler line of the reference triangle.〔 〔 The center of similitude of the orthic and tangential triangles is also on the Euler line.〔.〕〔 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「Euler line」の詳細全文を読む スポンサード リンク
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