2011 AMC 12B Problem 20

Triangle ABC has AB = 13, BC = 14, and AC = 15. The points D, E, and F are the midpoints of \overline{AB}, \overline{BC}, and \overline{AC} respectively. Let X \neq E be the intersection of the circumcircles of \Delta BDE and \Delta CEF. What is XA + XB + XC?

\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}

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