ISSN 1534-1178 Characterizations of an Infinite Set of Archimedean Circles

Abstract. For an arbelos with the two inner circles touching at a point O, we give necessary and sufficient conditions that a circle passing through O is Archimedean. Consider an arbelos with two inner circles α and β with radii a and b respec-tively touching externally at a point O. A circle of radius rA = ab/(a+b) is called Archimedean. In [3], we have constructed three infinite sets of Archimedean cir-cles. One of these consists of circles passing through the point O. In this note we give some characterizations of Archimedean circles passing through O. We set up a rectangular coordinate system with origin O and the positive x-axis along a diameter OA of α (see Figure 1)