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Area inequality - ILL 1990 GRE1

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September 18, 2010
geometryinequalities

Problem Statement

Point OO is interior to triangle ABCABC. Through OO, draw three lines DE∥BC,FG∥CADE \parallel BC, FG \parallel CA, and HI∥ABHI \parallel AB, where D,GD, G are on ABAB, I,FI, F are on BCBC and E,HE, H are on CACA. Denote by S1S_1 the area of hexagon DGHEFIDGHEFI, and S2S_2 the area of triangle ABCABC. Prove that S1≥23S2.S_1 \geq \frac 23 S_2.
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