OMM 2006 Problem 2

Let $ABC$ be a right triangle with a right angle at $A$ , such that $AB < AC$ . Let $M$ be the midpoint of $BC$ and $D$ the intersection of $AC$ with the perpendicular on $BC$ passing through $M$ . Let $E$ be the intersection of the parallel to $AC$ that passes through $M$ , with the perpendicular on $BD$ passing through $B$ . Show that the triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC = 60^o$ .