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Sets That Contain Their Circle Centers

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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 non-collinear points from S, the center of the circle through those three points is also an element of S. A problem appearing on the Macalester College Problem of the Week website stated 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 show how to prove a stronger assertion, namely that the one closed circle-center set is the entire plane, or equivalently that every circle-center set is dense in the plane. The step-by-step journey proceeds using elementary geometry for the most part, with a dash of plane topology thrown in.

Document Type: Research Article

Publication date: November 1, 2008

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