I would try to use Voronoi domains. I guess there are some algorithms which produce it. Then you only need to check the vertexes of Voronoi domains inside of $C_{n+1}$ and the points of intersections of the boundary of Voronoi domains and boundary of $C_{n+1}$.
In case of varying radii, you can still use modified Voronoi domains; $$V_i=\{\\,x\in\mathbb R^2\mid f_i(x)\le f_j(x)\ \text{for all}\ j\\,\}.$$ where $f_i(x)$ is the power of the point $x$ with respect to the corresponding circle $C_i$.