Density of circles centered on like circumference

[Loren Booda]

Imagine a circle of given radius. Construct all circles of equivalent radii whose centers constitute that initial circumference. Can you derive a probability density that describes the overall distribution of points from those resultant circles?

[Tron3k]

Unless I'm misinterpreting you, I would think that every point covered by these many circles except for the center point would have been overlapped exactly twice.