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Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.
In [Er81d] Erdős proved that $\gg n$ many circles is possible, and that there cannot be more than $n(n-1)$ many circles. Elekes [El84] has a simple construction of a set with $\gg n^{3/2}$ such circles. This may be the correct order of magnitude.