OMCPLP 2018 Problem 2

In a triangle $ABC$ , right in $A$ and isosceles, let $D$ be a point on the side $AC$ ( $A \ne D \ne C$ ) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$ , $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$ . Prove that the quadrilateral $ARQP$ is a square.