let $A, B$ be two points of the set such that all other points of the set lie on one side of the line $AB$. (e.g, take two consecutive vertices of the convex hull.) consider all the $2n+1$ circles through $A, B$ and the other points. the intersections of interiors of these circles with the halfplane on which the points lie can be ordered: $I_1 \subset I_2 \subset ... \subset I_{2n+1}$. the circle whose interior is $I_{n+1}$ contains exactly $n$ points in its interior.

It is a well known problem. It has even appeared in Mathematical Olympiad Challenge, by Titu Andreescu and Razvan Gelca.

it was also proposed for the 1995 IMO i think