There is clearly some non-degeneracy condition intended here - probably either that not all the points are on a line, or the stronger condition that no three points are on a line.

This was resolved by Elliott [El67], who proved that (assuming not all points are on a circle or a line) that, provided $n>393$, the points determine at least $\binom{n-1}{2}$ distinct circles.

The problem appears to remain open for small $n$. Segre observed that projecting a cube onto a plane shows that the lower bound $\binom{n-1}{2}$ is false for $n=8$.

See also [104] and [831].