KöMaL 2017 B4908

Let $C$ denote an arbitrary point on the circumference of a circle of diameter $AB$. Let $T$ be the perpendicular projection of point $C$ onto the line segment $AB$. Draw the circle of centre $C$ that passes through $T$. Let the intersections of the two circles be $P$ and $Q$. Prove that line $PQ$ bisects the line segment $CT$.