Ubiquitous Archimedean Circles of the Collinear Arbelos

For a point O on the segment AB in the plane, the area surrounded by the three semicircles with diameters AO, BO and AB erected on the same side is called an arbelos. It has lots of unexpected but interesting properties (for an extensive reference see [1]). The radical axis of the inner semicircles divides the arbelos into two curvilinear triangles with congruent incircles called the twin circles of Archimedes. Circles congruent to those circles are said to be Archimedean. In this paper we generalize the arbelos and the Archimedean circles, and show the existence of the generalized Archimedean circles covering the plane, which is a generalization of the ubiquitous Archimedean circles of the arbelos in [4].