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prove that xyz=x+y+z+2

Source: ItaMO 2004, p6

March 4, 2012
geometry proposedgeometry

Problem Statement

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