19. Circles with radii $R$ and $r$ touch each other internally. Find the side of a regular triangle, one vertex of which coincides with the point of tangency, and the other two lie on different given circles.
19. Let the side of the triangle be $x$ and the sides emanating from the common point of the circles form angles $\alpha$ and $\beta$ with the line passing through the centers, where $\alpha \pm \beta=60^{\circ}$; then $\cos \alpha=\frac{x}{2 R}, \cos \beta=\frac{x}{2 r}$ (or vice versa). Finding $\sin \alpha$ and $\sin \beta$, from the equation $\cos (\alpha \pm \beta)=$ $=\frac{1}{2}$ we get

$$
x=\frac{R_{r} \sqrt{ } 3}{\sqrt{R^{2}+r^{2}-R r}}
$$