IrMO 1990 Problem 5

Let $ABC$ be a right-angled triangle with right-angle at $A$ . Let $X$ be the foot of the perpendicular from $A$ to $BC$ , and $Y$ the mid-point of $XC$ . Let $AB$ be extended to $D$ so that $|AB|=|BD|$ . Prove that $DX$ is perpendicular to $AY$ .