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AMxCM + BMxDM >= \sqrt{ABxBCxCDxDA} if <AMB =<ADM+<BCM ...

Source: Ukrainian Geometry Olympiad 2020, XI p5

April 27, 2020

geometryanglesgeometric inequality

Problem Statement

Inside the convex quadrilateral

ABCD

there is a point

M

such that

\angle AMB = \angle ADM + \angle BCM

and

\angle AMD = \angle ABM + \angle DCM

. Prove that

AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}