More on twin circles of the skewed arbelos (Q2874407)

The arbelos configuration of Archimedes consists of three collinear points \(A\), \(O\), \(B\), three semicircles \(\alpha\), \(\beta\), \(\gamma\) having diameters \(AO\), \(OB\), \(AB\) and drawn on the same side of \(AB\), and a line \(OI\) perpendicular to \(AB\) and meeting \(\gamma\) at \(I\). The curvilinear triangle bounded by the line segment \(OI\) and the circular arcs \(\gamma\) and \(\alpha\) and the curvilinear triangle bounded by the line segment \(OI\) and the circular arcs \(\gamma\) and \(\beta\) were shown, by Archimedes, to have incircles of equal radii. Recently, other circles in the configuration that have equal radius have been discovered, as done by \textit{C. W. Dodge} et al. in [Math. Mag. 72, No. 3, 202--213 (1999; Zbl 1015.51007)]. Such circles, often coming in pairs, are now referred to as Archimedean. In [Forum Geom. 4, 229--251 (2004; Zbl 1070.51004)], the author and \textit{M. Watanabe} extended the arbelos configuration by adding a circle that touches \(\alpha\) and \(\beta\), and discovered other pairs of congruent circles.NEWLINENEWLINEThe paper under review considers again the extended configuration described above, and shows that two of the pairs of Archimedean circles that were discovered in the afore-mentioned paper of C. W. Dodge et al. can be viewed as special cases of more general pairs.NEWLINENEWLINEThis paper is a continuation of a series of papers that the author has written, jointly with others, in the same journal in the past several years. It is one more manifestation of the richness of the arbelos configuration.