IT-CESENATICO 2004 Problem 6

Let $P$ be a point inside a triangle $ABC$ . Lines $AP,BP,CP$ meet the opposite sides of the triangle at points $A',B',C'$ respectively. Denote $x =\frac{AP}{PA'}, y = \frac{BP}{PB'}$ and $z = \frac{CP}{PC'}$ . Prove that $xyz = x+y+z+2$ .