AT-OEMO-REG 2014 Problem 4

For a point $P$ in the interior of a triangle $ABC$ let $D$ be the intersection of $AP$ with $BC$ , let $E$ be the intersection of $BP$ with $AC$ and let $F$ be the intersection of $CP$ with $AB$ .Furthermore let $Q$ and $R$ be the intersections of the parallel to $AB$ through $P$ with the sides $AC$ and $BC$ , respectively. Likewise, let $S$ and $T$ be the intersections of the parallel to $BC$ through $P$ with the sides $AB$ and $AC$ , respectively.In a given triangle $ABC$ , determine all points $P$ for which the triangles $PRD$ , $PEQ$ and $PTE$ have the same area.