The circum-orthic triangle Δ A' B' C' is the circumcevian triangle of a triangle Δ A B C with respect to the orthocenter H. The circum-orthic triangle of a triangle Δ A B C is also the image of the orthic triangle in the homothecy centered at the orthocenter H of Δ A B C and having similitude ratio 2. It has trilinear vertex matrix [-a y z | (b y + c z) z | (b y + c z) y (c z + a x) z | -b z x | (c z + a x) x (a x + b y) y | (a x + b y) x | -c x y], where x = cos A, y = cos B, and z = cos C.

circumcevian triangle | medial triangle