The orthic inconic of a triangle is the inconic with inconic parameters x:y:z = cos A:cos B:cos C. It has trilinear equation a^2 S_A^2 α^2 - 2a b S_A S_B βα - 2a c S_A S_C γα + b^2 S_B^2 β^2 + c^2 S_C^2 γ^2 - 2b c S_B S_C βγ = 0 (P. Moses, pers. comm., Feb. 8, 2005), where S_A, S_B, and S_C are Conway triangle notation. It is an ellipse for acute triangles and a hyperbola for obtuse triangles.

Brianchon point | inconic | inellipse | MacBeath circumconic | orthic triangle