2002 AMC 12B Problem 23

In \triangle ABC, we have AB = 1 and AC = 2. Side \overline{BC} and the median from A to \overline{BC} have the same length. What is BC?

\mathrm{(A)}\ \frac{1+\sqrt{2}}{2} \qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2 \qquad\mathrm{(C)}\ \sqrt{2} \qquad\mathrm{(D)}\ \frac 32 \qquad\mathrm{(E)}\ \sqrt{3}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.