The inscribed circle ω of the triangle ABC touches its sides BC,CA and AB at points A1,B1 and C1, respectively. Let S be the intersection point of lines passing through points B and C and parallel to A1C1 and A1B1 respectively, A0 be the foot of the perpendicular drawn from point A1 on B1C1, G1 be the centroid of triangle A1B1C1, P be the intersection point of the ray G1A0 with ω. Prove that points S,A1, and P lie on a straight line. geometrycollinearincircleCentroid