PAMO 2012 Problem 1

$AB$ is a chord (not a diameter) of a circle with centre $O$ . Let $T$ be a point on segment $OB$ . The line through $T$ perpendicular to $OB$ meets $AB$ at $C$ and the circle at $D$ and $E$ . Denote by $S$ the orthogonal projection of $T$ onto $AB$ . Prove that $AS \cdot BC = TE \cdot TD$ .