AMC 12 2019b Problem 25

Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$?

$\textbf{(A) } 27 \textbf{(B) } 16\sqrt3 \textbf{(C) } 12+10\sqrt3 \textbf{(D) } 9+12\sqrt3 \textbf{(E) } 30$