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An Outstanding Inequality in an arbitrary hexagon, ZIMO - 09

Source: International Zhautykov Olympiad 2009, day 1, problem 3.

January 17, 2009
inequalitiesgeometrytrigonometrytrig identitiesLaw of Cosinesgeometry proposed

Problem Statement

For a convex hexagon ABCDEF ABCDEF with an area S S, prove that: AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S