Given an obtuse triangle, the polar circle has center at the orthocenter H. Call H_i the feet. Then the square of the radius r is given by r^2 | = | (H A)^_·(H H_A)^_ | = | (H B)^_·(H H_B)^_ | = | (H C)^_·(H H_C)^_ | = | -4 R^2 cos A cos B cos C | = | 4R^2 - 1/2(a^2 + b^2 + c^2), where R is the circumradius, A, B, and C are the angles, and a, b, and c are the corresponding side lengths. It is the anticomplement of the de Longchamps circle.

coaxal system | inversion pole | orthocentric system | polar | radical line