1. In $\triangle \mathrm{ABC}$, $\angle \mathrm{A}$ is twice $\angle \mathrm{B}$, given sides $\mathrm{b}$ and $c$, find side $a$.
Point $D$ and divide it into two parts with the ratio $b: c$. Secondly, since $\triangle A C D$ and $\triangle A B C$ are similar, we have,
$$
\frac{A C}{C D}=\frac{B C}{A C} \text { or written as } \frac{b}{\frac{a b}{b+c}}=\frac{a}{b} \text {. }
$$

Therefore, $\mathrm{a}=\sqrt{b^{2}+b c}$.