The Euler triangle of a triangle is the triangle whose vertices are the midpoints of the segments joining the orthocenter with the respective vertices. The vertices of the triangle are known as the Euler points, and lie on the nine-point circle. The Euler triangle is congruent and homothetic to the medial triangle and perspective to the orthic triangle (Kimberling 1998, p. 158).
It has trilinear vertex matrix
where , , and .
The following table gives the centers of the Euler triangle in terms of the centers of the reference triangle for Kimberling centers with .
center of Euler triangle | center of reference triangle | ||
incenter | midpoint of and | ||
triangle centroid | midpoint of and | ||
circumcenter | nine-point center | ||
orthocenter | orthocenter | ||
nine-point center | midpoint of and | ||
Nagel point | Fuhrmann center | ||
de Longchamps point | circumcenter | ||
perspector of abc and orthic-of-orthic triangle | -Ceva conjugate of | ||
homothetic center of orthic and tangential triangles | point Cheleb II | ||
Bevan point | Spieker center | ||
symmedian point of the anticomplementary triangle | reflection of in | ||
center of Jerabek hyperbola | |||
Tarry point | center of Kiepert hyperbola | ||
Steiner point | Kiepert antipode | ||
anticomplement of Feuerbach point | Feuerbach antipode |
A spherical triangle is sometimes also called Euler's triangle.