OPM 2008 Problem 5

Let $ABC$ be a right-angled triangle in $A$ such that $AB<AC$ . Let $M$ be the midpoint of $BC$ and let $D$ be the intersection of $AC$ with the perpendicular line to $BC$ which passes through $M$ . Let $E$ be the intersection point of the parallel line to $AC$ which passes through $M$ with the perpendicular line to $BD$ which passes through $B$ . Prove that triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC=60^{\circ}$ .