RO-TST 2014 Problem 14

Let $\triangle ABC$ be an acute triangle of circumcentre $O$ . Let the tangents to the circumcircle of $\triangle ABC$ in points $B$ and $C$ meet at point $P$ . The circle of centre $P$ and radius $PB=PC$ meets the internal angle bisector of $\angle BAC$ inside $\triangle ABC$ at point $S$ , and $OS \cap BC = D$ . The projections of $S$ on $AC$ and $AB$ respectively are $E$ and $F$ . Prove that $AD$ , $BE$ and $CF$ are concurrent.

Author: Cosmin Pohoata