In a triangle $ABC$, the altitude from $A$ meets the circumcircle again at $T$ . Let $O$ be the circumcenter. The lines $OA$ and $OT$ intersect the side $BC$ at $Q$ and $M$, respectively. Prove that
\[\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .\]