# Circles (A_{5a}) and (A_{5b}): The cousin circles

### Definition

In the smaller arbelos semicircles (O_{1}) and (O_{2}) make new arbeloi similar to the
original. The greater semicircle in the smaller of these arbeloi and the smaller
semicircle in the greater of these arbeloi are Archimedean [Dodge et al 1999].

### Properties

- The tangent from A to (O
_{2}) is tangent to (A_{5a}),
the tangent from B to (O_{1}) is tangent to (A_{5b}). [Dodge
et al 1999]
- The circular hull of (A
_{1a}) and (A_{2a}) is tangent to(A_{5a})
and (A_{5b}). By this we have in fact a *Powerian pair*. [van
Lamoen, 2007]

### Generalization

The first property mentioned can be extended to a property of the infinite
set of Archimedean circles through C. This generalization is: A circle through C
not tangent to (O_{1}) is Archimedean if and only if the two common
tangents of this circle and (O_{1}) meet on (O_{2}) (here
(O_{1}) and (O_{2}) are taken as complete circles). [Okumura
and Watanabe 2007]

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