Delaunay to Voronoi ... and back? Learning about Voronoi Diagrams, one quickly finds out that Delaunay Triangulations are clearly the easiest way to generate them from a set of points.

How about the other way around? Given a subdivision of (2D) space, how to determine unknown generators? Are there any common ways to do this?

Things I could think of were, assuming an underlying Delaunay triangulation, equal distances to Voronoi Vertices/"Delaunay Circumcenters", resulting in a system of quadratic equations.

Or searching for circles in the graph given by the Voronoi Vertices, and determining the center of the resulting polygons. 

There is also a relationship to k-means clustering, as clusters defined by k-means, being the generators, yield a Voronoi Diagram in turn. This would leave me having to determine k and populate the regions with elements that are precisely not equidistant to two means.

I feel likely to be missing something very obvious and I would be happy for any direction in which to get started.
Unfortunately, you cannot generally reconstruct the generators from Voronoi polygons. Look at the four simple examples below: all have the same two Voronoi cells but different placement of the generators.