SHARYGIN 2012 Problem 92

Three parallel lines passing through the vertices $A, B$ , and $C$ of triangle $ABC$ meet its circumcircle again at points $A_1, B_1$ , and $C_1$ respectively. Points $A_2, B_2$ , and $C_2$ are the reflections of points $A_1, B_1$ , and $C_1$ in $BC, CA$ , and $AB$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ are concurrent.

(D.Shvetsov, A.Zaslavsky)