How to segment a group of symmetric points

Question

I have a closed shape represented by N points in a 2D space, and I know for sure that the points have some sort of symmetry. Given the N points, how can we segment them into two symmetric sets?

Number of points is always even
Points are always connected to form a closed shape

** [EDIT]
Example

NeitherNor · Answer

The author confirmed my question in the comments that it is allowed to choose one of the two sets to be empty. Trivially, the empty set is symmetric. By assumption, its complement (the whole set) is symmetric, too.
Thus, the simplest answer to the question is: segment the points such that one of the two sets is empty!

How to segment a group of symmetric points

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