Given the circle $C_{1}$, midpoint $O_{1}(0,0)$ and radius $r$. Given the circle $C_{2}$, midpoint $O_{1}(r-s,0)$ and radius $s$. Point $A(r,0)$. Point of tangency $T(r-s+s\cos \alpha,s\sin \alpha)$. Equation of the tangent line $y\sin \alpha=(-x+r-s)\cos \alpha +s$. This line cuts the circle $C_{1}$ in the points $B$ and $C$: $BC=4\sqrt{r-s} \cdot \cos \frac{\alpha}{2} \cdot \sqrt{r\sin^{2}\frac{\alpha}{2}+s\cos^{2}\frac{\alpha}{2}}$. Also calculating $CA$ and $AB$, then $\frac{BC}{BC+CA+AB}=\frac{\sqrt{r(r-s)}-(r-s)}{s}$.