![]() In a right triangle, the Euler line coincides with the median to the hypotenuse-that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. The locus of the centroids of equilateral triangles inscribed in a given triangle is formed by two lines perpendicular to the given triangle's Euler line. Relation to inscribed equilateral triangles Moreover, the Euler line is parallel to an acute triangle's side BC if and only if : p.173 tan B tan C = 3. A proof of the fact that the circumcenter O O, the centroid G G and the orthocenter H H are collinear relies on free vectors. 447 : p.104, #211, p.242, #346 The center of similitude of the orthic and tangential triangles is also on the Euler line. The circumcenter of the tangential triangle lies on the Euler line of the reference triangle. ![]() The tangential triangle of a reference triangle is tangent to the latter's circumcircle at the reference triangle's vertices. 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. Other notable points that lie on the Euler line include the de Longchamps point, the Schiffler point, the Exeter point, and the Gossard perspector. 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. This property is also true for another triangle center, the nine-point center, although it had not been defined in Euler's time. Triangle centers on the Euler line Individual centers Įuler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear. ![]() The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron. It is a central line of the triangle, and 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. In geometry, the Euler line, named after Leonhard Euler ( / ˈ ɔɪ l ər/), is a line determined from any triangle that is not equilateral. Perpendicular lines from the side midpoints (intersect at the circumcenter) ![]()
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