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Prove that S{AQC}/S{CMT} = (sin B / cos C)^2

Source: Mediterranean MO 2004

October 31, 2010
geometrycircumcircletrigonometrygeometry proposed

Problem Statement

In a triangle ABCABC, the altitude from AA meets the circumcircle again at TT . Let OO be the circumcenter. The lines OAOA and OTOT intersect the side BCBC at QQ and MM, respectively. Prove that SAQCSCMT=(sin⁡Bcos⁡C)2.\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .
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