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\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}, parallelograms inscribed a convex ABCD

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p3

January 27, 2020
geometryparallelogramconvex quadrilateralarea of a trianglegeometric inequality

Problem Statement

Let PP be a point inside a convex quadrilateral ABCDABCD. Points K,L,M,NK,L,M,N are given on the sides AB,BC,CD,DAAB,BC,CD,DA respectively such that PKBLPKBL and PMDNPMDN are parallelograms. Let S,S1S,S_1, and S2S_2 be the areas of ABCD,PNAKABCD, PNAK, and PLCMPLCM. Prove that S≥S1+S2\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}.
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