BALKAN 1990 Problem 3

Let $ABC$ be an acute triangle and let $A_{1}, B_{1}, C_{1}$ be the feet of its altitudes. The incircle of the triangle $A_{1}B_{1}C_{1}$ touches its sides at the points $A_{2}, B_{2}, C_{2}$ . Prove that the Euler lines of triangles $ABC$ and $A_{2}B_{2}C_{2}$ coincide.