Picking Equal Areas

$ABCD$ is a right trapezium with $\angle DAB = \angle ABC = 90^\circ$ , $AB = 12$ , $BC = 32$ and $AD = 16$ . $E$ is a point on $DC$ such that $[ABED] = [BEC]$ . What is the value of $BE^2$ ?

Details and assumptions

$[PQRS]$ denotes the area of $PQRS$ .

A trapezium has a pair of parallel sides.