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 I J_1 as the diameter has P as its center, where P is the intersection of A_1 J_1 with the circumcircle of Δ A_1 A_2 A_3, and passes through A_2 and A_3. This circle has radius r = 1/2 a_1 sec(1/2 α_1) = 2R sin(1/2 α_1). It arises because I J_1 J_2 J_3 forms an orthocentric system.

circumcircle | excenter | excenter-excenter circle | incenter | orthocentric system