18. Let $A B C$ be an equilateral triangle. Let $\overrightarrow{A B}$ be extended to a point $D$ such that $B$ is the midpoint of $\overline{A D}$. A variable point $E$ is taken on the same plane such that $D E=A B$. If the distance between $C$ and $E$ is as large as possible, what is $\angle B E D$ ?
18. $15^{\circ}$
To make $C$ and $E$ as far as possible, $C, D, E$ must be collinear in that order.
With $\angle A B C=60^{\circ}$, we have $\angle C B D=120^{\circ}$. Since $B C=B D$, we then have $\angle C D B=\frac{1}{2}\left(180^{\circ}-120^{\circ}\right)=30^{\circ}$. Finally, since $B D=D E$, we have $\angle B E D=\frac{1}{2} \cdot 30^{\circ}=15^{\circ}$.