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Centroids form Equilateral Triangle

Source: 2019 AMC 12B #25

February 14, 2019
2019 AMC 12BAMCAMC 12AMC 12 B

Problem Statement

Let ABCDABCD be a convex quadrilateral with BC=2BC=2 and CD=6.CD=6. Suppose that the centroids of △ABC,△BCD,\triangle ABC,\triangle BCD, and △ACD\triangle ACD form the vertices of an equilateral triangle. What is the maximum possible value of the area of ABCDABCD?
<spanclass=′latex−bold′>(A)</span>27<spanclass=′latex−bold′>(B)</span>163<spanclass=′latex−bold′>(C)</span>12+103<spanclass=′latex−bold′>(D)</span>9+123<spanclass=′latex−bold′>(E)</span>30<span class='latex-bold'>(A) </span> 27 \qquad<span class='latex-bold'>(B) </span> 16\sqrt3 \qquad<span class='latex-bold'>(C) </span> 12+10\sqrt3 \qquad<span class='latex-bold'>(D) </span> 9+12\sqrt3 \qquad<span class='latex-bold'>(E) </span> 30
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