TUYMAADA 2007 Problem 3

$AA_{1}$ , $BB_{1}$ , $CC_{1}$ are altitudes of an acute triangle $ABC$ . A circle passing through $A_{1}$ and $B_{1}$ touches the arc $AB$ of its circumcircle at $C_{2}$ . The points $A_{2}$ , $B_{2}$ are defined similarly. Prove that the lines $AA_{2}$ , $BB_{2}$ , $CC_{2}$ are concurrent.