In $\triangle BAC$, $\angle BAC=30^\circ$, $AB=12$, $AC=8$. Points $D,E$ lie on $\overline{AB},\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?

1. Reflect $B$ in $AC$ to $B'$ and $C$ in $AB$ to $C'$. Then $BE+DE+CD=B'E+ED+DC'\ge B'C'$.
2. $AB'=12$, $AC'=8$, $\angle B'AC'=3\cdot30^\circ=90^\circ$, so $B'C'^2=12^2+8^2=208$.
3. $B'C'$ meets $AC$, $AB$ at points $E,D$ on the sides, so equality holds.
4. The minimum is $\sqrt{208}=\boxed{4\sqrt{13}}$.