The hidden square of the incircle

Let $\triangle ABC$ be an acute triangle with $AB=13$, $BC=14$, $CA=15$, incenter $I$, and let $M$ be the midpoint of $BC$. The incircle of $\triangle ABC$ touches $AB$ at $F$ and $AC$ at $E$. Let $P$ be the intersection of the line through $F$ perpendicular to $AB$ and the line through $E$ perpendicular to $AC$. Compute the exact value of $PI^2+PM^2$.