Consider the feet on AB of the tangents to the arbelos incircle (O3) perpendicular to AB. The equilateral triangle built on these two points has third vertex F. Inversion in the circle with center F through C maps the arbelos circles to congruent circles (E), (E1), and (E2). The circle tangent to AB and to (O) and (O3) in their common point meets (E1) and (E2) in their "highest" points. The two circles with the radii to these highest points as their diameters are the Archimedean circles (A36a) and (A36b). [Danneels and Van Lamoen, 2007].
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