Question 3
Two circles $\omega,\Omega$ are concentric. The chord $AD$ of $\Omega$ is tangent to $\omega$. Choose $P$ in the interior of the smaller circular segment bounded by $AD$ and $\Omega$. The tangents from $P$ to $\omega$ meet the major arc $AD$ of $\Omega$ at $B,C$. Segments $BD,AC$ meet at $Q$. Prove that segment $PQ$ passes through the midpoint of $AD$.