Given a triangle Δ A_1 A_2 A_3, the points A_1, I, and J_1 lie on a line, where I is the incenter and J_1 is the excenter corresponding to A_1. Furthermore, the circle with J_2 J_3 as the diameter has Q as its center, where P is the intersection of A_1 J_1 with the circumcircle of A_1 A_2 A_3 and Q is the point opposite P on the circumcircle. The circle with diameter J_2 J_3 also passes through A_2 and A_3 and has radius r = 1/2 a_1 csc(1/2 α_1) = 2R cos(1/2 α_1). It arises because the points I, J_1, J_2, and J_3 form an orthocentric system.

excenter | incenter-excenter circle | orthocentric system