Given a triangle $ABC$ with vertices at $A(2, 3)$, $B(8, 3)$, and $C(5, 9)$, find the area of the triangle. Then inscribe a rectangle $DEFG$ within the triangle such that $D$ is on $AB$, $E$ is on $BC$, $F$ is on $CA$, and $G$ is on $AC$. The lengths of $DE$ and $FG$ are equal. Calculate the area of the rectangle $DEFG$. Let's think step by step and output the final answer within \boxed{}.
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