Let P be a point in the line GGe (G centroid, Ge Gergonne ppoint), A1B1C1 its cevian triangle. Draw a circle tangent to BC at A1 and to the circumcircle at A2, A2 outside the arc of BC containing A1. Define analogously B2, C2. Then A1A2, B1B2, C1C2 concur at a ppoint in OI (O circumcenter, I incenter).