David Graham Searby, On three circles,
Forum Geometricorum, 9 (2009) 181--193.

Abstract:  The classical Three-Circle Problem of Apollonius requires the construction of a fourth circle tangent to three given circles in the Euclidean plane. For circles in general position this may admit as many as eight solutions or even no solutions at all.  Clearly, an ``experimental" approach is unlikely to solve the problem,  but, surprisingly, it leads to a more general theorem. Here we consider the case of a chain of circles which, starting from an arbitrary point on one of the three given circles defines (uniquely, if one is careful) a tangent circle at this point and a tangency point on another  of the given circles. Taking this new point as a base we construct a circle tangent to the second circle at this point and to the third circle, and repeat the construction  cyclically. For any choice of the three starting circles, the tangency points are concyclic and the chain can contain at most six circles. The figure reveals unexpected connections with many classical theorems of projective geometry, and it admits the Three-Circle Problem of Apollonius as a particular case.

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