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? $\textbf{(A) }100\sqrt{2}\qquad$ $\textbf{(B) }100\sqrt{3}\qquad$ $\textbf{(C) }200\qquad$ $\textbf{(D) }200\sqrt{2}\qquad$ $\textbf{(E) }200\sqrt{3}$