IMO 2008 Problem 1

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ . The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and $A_{2}$ . Similarly, define the points $B_{1}$ , $B_{2}$ , $C_{1}$ and $C_{2}$ .

Prove that the six points $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ , $C_{1}$ and $C_{2}$ are concyclic.

Author: Andrey Gavrilyuk, Russia