Sets That Contain Their Circle Centers

The terminology needs a brief explanation. A set S ⊂ R2 is said to contain its circle centers if, for any three non-collinear points from 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 below) 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 naturally turned at that point to how much could be said about sets that contain their circle centers. The entire plane S = R2 is a perfectly reasonable example of such a set; less trivially, the set S = Q2, consisting of all points in the plane with rational numbers as coordinates, contains its circle centers (see Exercise 3 below). Are there lots of sets with this property, or only a few? It’s clearly cheating to put all the points of S on a single line, or else there are no circle centers to form. This motivates the following definition: