2012 APMO 第 4 题

Let $P$ be a point in the interior of a triangle $ABC$ , and let $D, E, F$ be the point of intersection of the line $AP$ and the side $BC$ of the triangle, of the line $BP$ and the side $CA$ , and of the line $CP$ and the side $AB$ , respectively. Prove that the area of the triangle $ABC$ must be $6$ if the area of each of the triangles $PFA, PDB$ and $PEC$ is $1$ .

设$P$为三角形$ABC$内部的一点，设$D、E、F$分别为三角形$AP$与边$BC$的交点、$BP$与边$CA$的交点、$CP$与边$AB$的交点。证明如果$PFA、PDB$和$PEC$每个三角形的面积都是$1$，那么三角形$ABC$的面积一定是$6$。