PE-MO 2021 Problem 3

Let $M,N,P$ be points in the sides $BC,AC,AB$ of $\triangle ABC$ respectively. The quadrilateral $MCNP$ has an incircle of radius $r$ , if the incircles of $\triangle BPM$ and $\triangle ANP$ also have the radius $r$ . Prove that $$AP\cdot MP=BP\cdot NP$$