ISL 1994 G5

A circle $C$ with center $O.$ and a line $L$ which does not touch circle $C.$ $OQ$ is perpendicular to $L,$ $Q$ is on $L.$ $P$ is on $L,$ draw two tangents $L_1, L_2$ to circle $C.$ $QA, QB$ are perpendicular to $L_1, L_2$ respectively. ( $A$ on $L_1,$ $B$ on $L_2$ ). Prove that, line $AB$ intersect $QO$ at a fixed point.

Original formulation:

A line $l$ does not meet a circle $\omega$ with center $O.$ $E$ is the point on $l$ such that $OE$ is perpendicular to $l.$ $M$ is any point on $l$ other than $E.$ The tangents from $M$ to $\omega$ touch it at $A$ and $B.$ $C$ is the point on $MA$ such that $EC$ is perpendicular to $MA.$ $D$ is the point on $MB$ such that $ED$ is perpendicular to $MB.$ The line $CD$ cuts $OE$ at $F.$ Prove that the location of $F$ is independent of that of $M.$