Question 32
Let $ABC$ be an acute-angled and scalene triangle, $\omega$ its incircle and $\omega'$ the excircle relative to the vertex $A$. The circles $\omega$ and $\omega'$ are tangent to $BC$ at $P$ and $P'$ respectively. Let $\Gamma$ be the circumference passing through $B$ and $C$ that is tangent to $\omega$ in a point $Q$, and $\Gamma'$ be the circumference passing through $B$ and $C$ that is tangent to $\omega'$ in a point $Q'$. The lines $PQ$ and $P'Q'$ intersect in $N$. Prove that $AN$ is perpendicular to $BC$.