2014 AMC 12A Problem 20

In \triangle BAC, \angle BAC=40^\circ, AB=10, and AC=6. Points D and E lie on \overline{AB} and \overline{AC} respectively. What is the minimum possible value of BE+DE+CD?

\textbf{(A) }6\sqrt 3+3\qquad \textbf{(B) }\dfrac{27}2\qquad \textbf{(C) }8\sqrt 3\qquad \textbf{(D) }14\qquad \textbf{(E) }3\sqrt 3+9\qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.