Problem

Source: Iran TST 2012 -first day- problem 2

Tags: geometry, circumcircle, incenter, trigonometry, angle bisector, power of a point, config geo



Consider $\omega$ is circumcircle of an acute triangle $ABC$. $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$. Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$. Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$. Prove that $PE$ is bisector of angle $BPC$. Proposed by Mr.Etesami