OME 2023 Problem 2

Let $ABC$ be an acute scalene triangle with incenter $I$ and orthocenter $H$ . Let $M$ be the midpoint of $AB$ . On the line $AH$ we consider points $D$ and $E$ , such that the line $MD$ is parallel to $CI$ and $ME$ is perpendicular to $CI$ . Prove that $AE=DH$ .