12.46 The magnitude of the angle at the base of an isosceles triangle is equal to $\alpha$. For what value of $\alpha$ is the ratio of the lengths of the radii of the inscribed and circumscribed circles the greatest? What is this ratio?
12.46 Instruction. Let in triangle $ABC$ points $O$ and $O_{1}$ be the centers of the circumscribed and inscribed circles, and $R$ and $r$ the radii of these circles (Fig. 12.8). Express the ratio $\frac{r}{R}$ in terms of $\alpha$ and investigate the function $\frac{r}{R}=f(\alpha)=\sin 2 \alpha \tan \frac{\alpha}{2}$ for extrema.

Answer: $\alpha=\frac{\pi}{3} ; \frac{r}{R}=\frac{1}{2}$.