# Circle (A_{36a}) and (A_{36b}): Congruent inversion image
circles

### Definition

Consider the feet on AB of the tangents to the arbelos incircle (O_{3})
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), (E_{1}), and (E_{2}). The
circle tangent to AB and to (O) and (O_{3}) in their common point meets
(E_{1}) and (E_{2}) in their "highest" points. The two circles
with the radii to these highest points as their diameters are the Archimedean
circles (A_{36a}) and (A_{36b}). [Danneels and Van Lamoen,
2007].

### Properties:

- The circles (A
_{36a}) and (A_{36b}) are tangent to (A_{2}).
[Danneels and Van Lamoen, 2007]

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