INMO 2015 Problem 5

Let $A B C D$ be a convex quadrilateral. Let the diagonals $A C$ and $B D$ intersect in $P$. Let $P E, P F, P G$ and $P H$ be the altitudes from $P$ on to the sides $A B, B C, C D$ and $D A$ respectively. Show that $A B C D$ has an incircle if and only if

$$\frac{1}{P E}+\frac{1}{P G}=\frac{1}{P F}+\frac{1}{P H}$$