smc

Let $[ABC]$ denote the area of triangle $ABC$. $[AED]=[BEC]$, so $[ABD]=[AED]+[AEB]=[BEC]+[AEB]=[ABC]$. Since triangles $ABD$ and $ABC$ share a base, they also have the same height and thus $\overline{AB}||\overline{CD}$ and $△AEB\sim △CED$ with a ratio of $3:4$. $AE=\frac{3}{7}\times AC$, so $AE=\frac{3}{7}\times 14=6\boxed{6}$.