Question 20
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose $AP, BP, CP$ meet the sides $BC, CA, AB$ (or extensions thereof) in $D, E, F$, respectively. Suppose further that the areas of triangles $PBD, PCE, PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.