2001 IMO Shortlist Problems/G6

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Problem

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$ , $BP$ , $CP$ 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.