The Number of Halving Circles Federico Ardila Summary: The Number of Halving Circles Federico Ardila 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 n2 halving circles. Our starting point is the following problem, which appeared in the 1962 Chinese Mathematical Olympiad [7]. Problem 1. Prove that any set of 2n + 1 points in general position in the plane has a halving circle. For the rest of sections 1 and 2, n is a fixed positive integer and S signifies an arbitrary set of 2n + 1 points in general position in the plane. There are several solutions to Problem 1. One possible approach is the following. Let A and B be two consecutive vertices of the convex hull of S. We claim that some circle going through A and B is halving. All circles through A and B have their centers on the perpendicular bisector of the segment AB. Pick a point O on that lies on the same side of AB as S and is sufficiently far away from AB that the circle with Collections: Mathematics