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Recently, I solved an analytical problem on the number of possible regions in a similar figure:

enter image description here

I conducted numerical experiments on Mathematica to verify the results. Intersection points are easy to find and count.
But it turned out to be difficult to detect regions. The main thing, is to find polygons. There are special algorithms, and of course they can be implemented on Mathematica.

But I would like to use built-in features of Geometric Computation. However, I did not succeed (although I know these opportunities quite well)

Finally the next unusal code works well:

g = Graphics[{Thickness[0.01], Circle[], lines}, Background -> White];
numareas = Max@MorphologicalComponents[g] - 1;

But the question remains: is it possible to use native integrated capabilities of Geometric Computation?

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