The Number of Halving Circles

1. INTRODUCTION. We say that a set S of 2n + 1 points in the plane is in general position if no three of the points are collinear and no four are concyclic. We call a circle halving with respect to S if it has three points of S on its circumference, n − 1 points in its interior, and n − 1 in its exterior. The goal of this paper is to prove the following surprising fact: any set of 2n + 1 points in general position in the plane has exactly n 2 halving circles. Our starting point is the following problem, which appeared in the 1962 Chinese Mathematical Olympiad [7].