  A set S of 2n+1 points in the plane is said to be in general position if no
three points of S are collinear and no four are concyclic. A circle is called
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. We prove the following
surprising result: any set of 2n+1 points in general position in the plane has
exactly n^2 halving circles.
