AMC 12 2002b 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 $BC$?

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