Let P be a point, A1B1C1 its cevian triangle. Draw two circles tangent to BC at A1 and to the circumcircle at A2, A3; A2 outside the arc of BC containing A1, A3 inside. Define analogously B2, B3, C2, C3. Then A2A3, B2B3, C2C3 concur at Q; when P is the centroid Q is the circumcenter; when P is the incenter Q is the symmedian.