International Mathematical Olympiad 1994 Problem 2

$ABC$ is an isosceles triangle with $AB = AC$. Suppose that

1. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$;
2. $Q$ is an arbitrary point on the segment $BC$ different from $B$ and $C$;
3. $E$ 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 $QE = QF$.