Question 37
Let $O$ be the circumcenter of an acute-angled triangle $ABC$. Let $AH$ be the altitude of this triangle, $M$, $N$, $P$, $Q$ be the midpoints of the segments $AB$, $AC$, $BH$, $CH$, respectively.
Let $\omega_1$ and $\omega_2$ be the circumcircles of the triangles $AMN$ and $POQ$.
Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$.
Let $\omega_1$ and $\omega_2$ be the circumcircles of the triangles $AMN$ and $POQ$.
Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$.