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Area inequality in quadrilateral

Source: Austrian-Polish 2001, Problem 4

August 3, 2015
geometryinequalitiesquadrilateral

Problem Statement

Prove that if a,b,c,da,b,c,d are lengths of the successive sides of a quadrangle (not necessarily convex) with the area equal to SS, then the following inequality holds S≤12(ac+bd).S \leq \frac{1}{2}(ac+bd). For which quadrangles does the inequality become equality?
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