Question 11
We are given an acute-angled triangle $ABC$. Denote by $k$ the circle with diameter $AB$. A circle touching the bisector of the angle $BAC$ at the point $A$ and passing through the point $C$ meets the circle $k$ in a point $P$, $P \neq A$. Similarly, a circle touching the bisector of the angle $ABC$ at the point $B$ and passing through the point $C$ meets the circle $k$ in a point $Q$, $Q \neq B$. Prove that the lines $AQ$ and $BP$ intersect each other on the bisector of the angle $ACB$. (Peter Novotný)