Mathematical Olympiad Rioplatense

Consider a semicircle with diameter $AB$, center $O$ and radius $r$. Let $C$ be the point on segment $AB$ such that $AC=\frac{2r}{3}$. Line $l$ is perpendicular to $AB$ at $C$ and $D$ is the common point of $l$ and the semicircle. Let $H$ be the foot of the perpendicular from $O$ to $AD$ and $E$ the intersection of lines $CD$ and $OH$.

a) Express $AD$ as a function of $r$.

b) If $M$ and $N$ are the midpoints of $AE$ and $OD$ respectively, find the measure of angle $MHN$.