Question 49
Let $AA_{1}$, $BB_{1}$ and $CC_{1}$ be the altitudes of an acute $\triangle ABC$ ($A_{1} \in BC$, $B_{1} \in CA$ and $C_{1} \in AB$). Denote by $O$ the circumcenter of $\triangle ABC$, and by $H_{1}$ the orthocenter of $\triangle A_{1}B_{1}C_{1}$. Prove that the midpoint of the segment $OH_{1}$ coincides with the incenter of the triangle with vertices at the midpoints of the sides of $\triangle A_{1}B_{1}C_{1}$.