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IMO ShortList 1999, geometry problem 7

Source: IMO ShortList 1999, geometry problem 7

November 13, 2004
geometrycircumcirclereflectionsimilar trianglesquadrilateralIMO Shortlist

Problem Statement

The point MM is inside the convex quadrilateral ABCDABCD, such that MA = MC, \hspace{0,2cm} \widehat{AMB} = \widehat{MAD} + \widehat{MCD}   \textnormal{and}   \widehat{CMD} = \widehat{MCB} + \widehat{MAB}. Prove that AB⋅CM=BC⋅MDAB \cdot CM = BC \cdot MD and BM⋅AD=MA⋅CD.BM \cdot AD = MA \cdot CD.
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