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.