I need to find the radius of the circle on the Riemann sphere $S$ whose stereographic projection is $C(a;r)$, i.e. the circle with centre $a$ and radius $r$ in the complex plane.
I have observed that, if two diametrically opposite points on the circle $C(a;r)$ have also their preimage as two diametrically opposite points of the circle on $S$, then one can compute the required radius which is $\frac{1}{2}\sigma (|a|+r,|a|-r)$ (where $\sigma$ is the metric corresponding to stereographic projection). It is kind of obvious, but I am unable to prove this fact.