2. In a right-angled triangle, the legs are equal to $b$ and $c$, respectively. Find the length of the angle bisector of the right angle.



The above text has been translated into English, preserving the original text's line breaks and format.
2. Let $\mathrm{AD}$ be the external angle bisector of the right angle $\mathrm{A}$ in $\triangle \mathrm{ABC}$, and $\mathrm{DE} \parallel \mathrm{AC}$ (Figure 2). Given that $\angle \mathrm{DAE}=\frac{\pi}{4}$, then $\mathrm{AE}=\mathrm{DE}=\frac{x}{\sqrt{2}}$, where the solution $x=\mathrm{AD}$ is the required length.

Next, $\frac{\mathrm{ED}}{\mathrm{AB}}=\frac{\mathrm{CE}}{\mathrm{CA}}$, or written as $\frac{\sqrt{2}}{\mathrm{c}}=-\frac{\mathrm{x}}{\mathrm{b}}$. From this, we get $\quad x=\frac{b c \sqrt{2}}{b+c}$.