IMO 1994 Problem 2

Let $ABC$ be an isosceles triangle with $AB = AC$ . $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$ . $Q$ is an arbitrary point on $BC$ different from $B$ and $C$ . $E$ lies on the line $AB$ and $F$ lies on the line $AC$ such that $E, Q, F$ are distinct and collinear. Prove that $OQ$ is perpendicular to $EF$ if and only if $QE = QF$ .