RO-TST 2008 Problem 11

Let $ABC$ be an acute triangle with orthocenter $H$ and let $X$ be an arbitrary point in its plane. The circle with diameter $HX$ intersects the lines $AH$ and $AX$ at $A_{1}$ and $A_{2}$ , respectively. Similarly, define $B_{1}$ , $B_{2}$ , $C_{1}$ , $C_{2}$ . Prove that the lines $A_{1}A_{2}$ , $B_{1}B_{2}$ , $C_{1}C_{2}$ are concurrent.