Question 64
Let $ABC$ be a triangle with $\angle BCA = 90^\circ$ and let $H$ be the foot of the altitude from $C$. Let $D$ be a point inside the triangle $BCH$ such that $CH$ bisects the segment $AD$. Let $P$ be the intersection of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$. The tangent from $P$ to $\omega$ touches it at $Q$. Show that the intersection point of the lines $CQ$ and $AD$ lies on $\omega$.