AHSME 1983 Problem 10

Segment $AB$ is both a diameter of a circle of radius $1$ and a side of an equilateral triangle $ABC$. The circle also intersects $AC$ and $BC$ at points $D$ and $E$, respectively. The length of $AE$ is $\textbf{(A)} \ \frac{3}{2} \textbf{(B)} \ \frac{5}{3} \textbf{(C)} \ \frac{\sqrt 3}{2} \textbf{(D)}\ \sqrt{3} \textbf{(E)}\ \frac{2+\sqrt 3}{2}$