Question 148
Let $ABC$ be a triangle with $\angle C = 90^\circ$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$.
Prove that the lines $CQ$ and $AD$ meet on $\omega$.
(IMO-2015 Shortlist, Problem G3)
Prove that the lines $CQ$ and $AD$ meet on $\omega$.
(IMO-2015 Shortlist, Problem G3)