2009 AMC 10A Problem 23

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?

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

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