MathNet 2014 Question 23

Let $ABC$ be an acute-angled triangle and let $O$ be its circumcenter. The tangents of the circumcircle $ABC$ at vertices $B$ and $C$ meet at $P$, the circle of radius $PB$ centered at $P$ meets the internal angle bisector of the angle $BAC$ at point $Q$ lying in the interior of the triangle $ABC$, and the lines $OQ$ and $BC$ meet at $D$. Finally, let $E$ and $F$ be the orthogonal projections of $Q$ on the lines $AC$ and $AB$, respectively. Prove that the lines $AD, BE$ and $CF$ are concurrent.
Cosmin Pohoăţă