Sets That Contain Their Circle Centers

Abstract

The terminology needs a brief explanation. A set S in the plane is said to contain its circle centers if, for any three noncollinear points in S, the center of the circle through those points is always in S; in other words, if S contains the vertices of any triangle, then it also contains the triangle’s circumcenter. Several solutions were quickly submitted, and two (outlined in Exercises 1 and 2) were posted on the website, along with some discussion. An especially interesting feature of this pair of solutions was that one did not use the assumption that no three points of S lie on a line, while the other did not use the assumption that S is finite! The discussion turned naturally at that point to how much could be said about sets that do contain their circle centers. The entire plane is a perfectly reasonable example of such a set; less trivially, the set Q of all points in the plane with rational coordinates contains its circle centers (see Exercise 3). Are there lots of sets with this property, or only a few? It’s clearly cheating to have all the points of S on one line, since then there are no circle centers. This motivates the following definition: A circle-center set is a set of points in the plane that is not a subset of any line and that contains all of its circle centers. In the subsequent discussion on the website, John Guilford conjectured that any circle-center set must be unbounded. (Note that the circumcenter of an obtuse triangle is actually outside the triangle, and so a circle-center set containing obtuse triangles may well have quite distant points.) Working on this conjecture eventually led to an even stronger discovery: there is essentially only one circle-center set. The reader might well be skeptical at this point, especially given the fact that we have already mentioned two circle-center sets, namely Q and the entire plane! To pin down what we mean by “essentially”, we need to review some basic topological terms about sets in the plane. An open disk is simply the inside of any circle. A set S is dense if every open disk contains at least one of the set’s points, or equivalently if for any point P in R, there is a sequence of points P1, P2, P3, . . . in S that converges to P . A closed set is a set S with the property that, for any sequence of points P1, P2, P3, . . . from S whose limit exists, the limit itself must be in S. Finally, the closure S of S is the set of all points we can obtain by taking limits of sequences of points from S, or equivalently the smallest closed set containing S. Using this terminology, the claim that there is essentially only one circle-center set can be made precise as follows: