FORUM GEOM ISSN 1534-1178 Some Identities Arising From Inversion of Pappus Chains in an Arbelos

Abstract. We consider the inversive images, with respect to the incircle of an arbelos, of the three Pappus chains associated with the arbelos, and establish some identities connecting the radii of the circles involved. In a previous work [1], we considered the three Pappus chains that can be drawn inside the arbelos and demonstrated some identities relating the radii of the circles in these chains. In Figure 1, the diameter AC of the left semicircle Ca is 2a, the diameter CB of the right semicircle Cb is 2b, and the diameter AB of the outer semicircle Cr is 2r, r = a+ b. The first circle Γ1 is common to all three chains and is the incircle of the arbelos. C BA Figure 1. The Pappus chains in an arbelos With reference to Figure 1, we denote by Γr, Γa and Γb the chains converging to C, A, B respectively. Table 1 gives the coordinates of the centers and the radii of the circles in the chains, referring to a Cartesian reference system with origin at C and x-axis along AB