Euler Triangle

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.