Square $ABCD$ has side length 30. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26.$ The centroids of $\triangle ABP,$ $\triangle BCP,$ $\triangle CDP,$ and $\triangle DAP$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral?

200
$200 \sqrt{3}$
$100 \sqrt{2}$
$200 \sqrt{2}$
$100 \sqrt{3}$

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