Question 38
For a point $P$ in the interior of a triangle $ABC$, let $D$ be the intersection of $AP$ with $BC$, $E$ the intersection of $BP$ with $AC$ and $F$ 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.
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.