The Macbeath circumconic is the dual conic to the Macbeath inconic, introduced in Dec. 2004 by P. Moses (Kimberling). It has circumconic parameters x:y:z = cos A:cos B:cos C, and so has trilinear equation (cos A)/α + (cos B)/β + (cos C)/γ = 0. Its center is the symmedian point K. When it is a circumellipse, it has area A = (2πsqrt(2)a b c sqrt(sec A sec B sec C))/(a^2 + b^2 + c^2)^(3/2) Δ.

circumconic | Macbeath inconic

Alexander Murray Macbeath