Let ABC be a triangle with points D1 # D2 on the side BC, points E1 + E2 on the side AC and points F1 # F2 on the side AB. Assume that points D1, D2, E1, E2, F1, F2 lie on a common circle. Prove that if the lines AD1, BE1, CF intersect in a common point, then the lines AD2, BE2, CF2 also intersect in a common point. Hint: Ceva's theorem and the power of a point with respect to a circle.