Perpendicular bisectors in a circle

Geometry Level 5

Let SS be a set of 31 31 equally spaced points on a circle centered at O O , and consider a uniformly random pair of distinct points A A and B B (A,BSA, B \in S). The probability that the perpendicular bisectors of OA OA and OB OB intersect strictly inside the circle can be expressed as mn \frac{m}{n} , where m,n m,n are relatively prime positive integers. Find m+n m+n .

This problem is posed by Muhammad A.

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