NL-NWO 2021 Problem 4

In triangle $ABC$ we have $\angle ACB = 90^o$ . The point $M$ is the midpoint of $AB$ . The line through $M$ parallel to $BC$ intersects $AC$ in $D$ . The midpoint of line segment $CD$ is $E$ . The lines $BD$ and $CM$ are perpendicular. (a) Prove that triangles $CME$ and $ABD$ are similar. (b) Prove that $EM$ and $AB$ are perpendicular.

Be aware: the figure is not drawn to scale.