A combinatorial result on points and circles on the plane

Let P n be a collection of n points on the plane. For a pair of points u and v P n let C(u,v) be the minimum number of points of P n contained in any circle contaning u and v. In this paper we prove the result that there exist two points u o and v o P n such that any circle containing u o and v o contains at least È(n-2)/60˘ elements of P n (other than u o and v o). We also prove that the average value of C(u,v) over all pairs {u, v} P n is ≥ È(n-2)/60˘. For the case when P n are the vertices of a convex polygon, we prove that there exist two vertices u, v of P n such that any circle containing them contains at least È(n-2)/4˘ elements of P n .