Circum-Orthic Triangle

The circum-orthic triangle is the circumcevian triangle of a triangle with respect to the orthocenter (Kimberling 1998, p. 163). The circum-orthic triangle of a triangle is also the image of the orthic triangle in the homothecy centered at the orthocenter of and having similitude ratio 2.

It has trilinear vertex matrix

where , , and .

Its area is

where is the area of .

The following table gives the centers of the circum-orthic triangle in terms of the centers of the reference triangle corresponding to Kimberling centers .

	center of circum-orthic triangle		center of reference triangle
	circumcenter		circumcenter
			anticomplement of
	Tarry point		Collings transform of
	Steiner point		Collings transform of
	focus of Kiepert parabola		Gibert point
	triangle centroid of the antipedal triangle of		triangle centroid of dual triangle of
	isogonal conjugate of		Napoleon crossdifference