Question 88
In triangle $ABC$ the midpoints of the sides $BC$, $CA$ and $AB$ are $D$, $F$, $E$ respectively. Let $M$ be the point where the internal bisector of the angle $\angle ADB$ meets the side $AB$, and $N$ the point where the internal bisector of the angle $\angle ADC$ meets the side $AC$. Furthermore, let $O$ be the intersection point of the lines $AD$ and $MN$, $P$ the intersection point of $AB$ and $FO$, and $R$ the intersection point of $AC$ and $EO$. Prove that $PR = AD$.