  Say that a subset S of the plane is a "circle-center set" if S is not a
subset of a line, and whenever we choose three noncollinear points from S, the
center of the unique circle through those three points is also an element of S.
A problem appearing on the Macalester College Problem of the Week website was
to prove that a finite set of points in the plane, no three lying on a common
line, cannot be a circle-center set. Various solutions to this problem that did
not use the full strength of the hypotheses appeared, and the conjecture was
subsequently made that every circle-center set is unbounded. In this article,
we prove a stronger assertion, namely that every circle-center set is dense in
the plane, or equivalently that the only closed circle-center set is the entire
plane. Along the way we show connections between our geometrical method of
proof and number theory, real analysis, and topology.
