OME 2017 Problem 6

In the triangle $ABC$ , the respective mid points of the sides $BC$ , $AB$ and $AC$ are $D$ , $E$ and $F$ . Let $M$ be the point where the internal bisector of the angle $\angle ADB$ intersects the side $AB$ , and $N$ the point where the internal bisector of the angle $\angle ADC$ intersects the side $AC$ . Also, let $O$ be the intersection point of $AD$ and $MN$ , $P$ the intersection point of $AB$ and $FO$ , and $R$ the intersection point of $AC$ and $EO$ . Prove that $PR=AD$ .