I would try to use [Voronoi domains][1]. 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][2] $x$ with respect to the corresponding circle $C_i$.


  [1]: http://en.wikipedia.org/wiki/Voronoi_diagram
  [2]: http://en.wikipedia.org/wiki/Power_of_a_point