NL-BXMO-TST 2010 Problem 1

Let $ABCD$ be a trapezoid with $AB // CD$ , $2|AB| = |CD|$ and $BD \perp BC$ . Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$ . Let $O$ be the intersection of $AM$ and $BD$ . Let $N$ be the intersection of $OE$ and $AB$ . (a) Prove that $ABMD$ is a rhombus. (b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$ .