MacBeath ellipse

The MacBeath inconic of a triangle is the inconic with parameters x:y:z = a^2 cos A:b^2 cos B:c^2 cos C. Its foci are the circumcenter O and the orthocenter H, giving the center as the nine-point center N. It is named after Macbeath, but had earlier been investigated by Serret. It was subsequently publicized by Gabriel-Marie. The Brianchon point is the isotomic conjugate of the circumcenter O, which is Kimberling center X_264 and has triangle center function α_264 = (sec A)/a^2.

inconic | MacBeath circumconic | MacBeath triangle