MEMO 2016 Problem 6

Let $ABC$ be a triangle for which $AB \neq AC$ . Points $K$ , $L$ , $M$ are the midpoints of the sides $BC$ , $CA$ , $AB$ . The incircle of $ABC$ with center $I$ is tangent to $BC$ in $D$ . A line passing through the midpoint of $ID$ perpendicular to $IK$ meets the line $LM$ in $P$ .

Prove that $\angle PIA = 90 ^{\circ}$ .