AMC 12 2009a Problem 20

Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$?

\textbf{(A)}\ \frac {9}{2} \textbf{(B)}\ \frac {50}{11} \textbf{(C)}\ \frac {21}{4} \textbf{(D)}\ \frac {17}{3} \textbf{(E)}\ 6