ILL 1990 Problem 21

Point $O$ is interior to triangle $ABC$ . Through $O$ , draw three lines $DE \parallel BC, FG \parallel CA$ , and $HI \parallel AB$ , where $D, G$ are on $AB$ , $I, F$ are on $BC$ and $E, H$ are on $CA$ . Denote by $S_1$ the area of hexagon $DGHEFI$ , and $S_2$ the area of triangle $ABC$ . Prove that $S_1 \geq \frac 23 S_2.$