Convex quadrilateral $ABCD$ has $AB=10$ and $CD=15$. Diagonals $AC$ and $BD$ intersect at $E$, $AC=20$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$?

1. $[ADE]=[BCE]$ forces $AB\parallel CD$, so $\triangle ABE\sim\triangle CDE$.
2. $AE:CE=AB:CD=10:15=2:3$, so $AE=\dfrac25(20)=\boxed8$.