SHARYGIN 2006 Problem 104

Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$ . The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$ . Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.