OMM 1992 Problem 6

$ABCD$ is a rectangle. $I$ is the midpoint of $CD$ . $BI$ meets $AC$ at $M$ . Show that the line $DM$ passes through the midpoint of $BC$ . $E$ is a point outside the rectangle such that $AE = BE$ and $\angle AEB = 90^o$ . If $BE = BC = x$ , show that $EM$ bisects $\angle AMB$ . Find the area of $AEBM$ in terms of $x$ .