A note on a problem of Apollonius

@article{FitzGerald1974ANO,
  title={A note on a problem of Apollonius},
  author={J. Fitz-Gerald},
  journal={Journal of Geometry},
  year={1974},
  volume={5},
  pages={15-26}
}
Degenerate cases of the problem of Apollonius, to construct a circle tangent to each of three given circles, are discussed and exhaustively classified for proper circles (finite and non-zero radius). Singular cases are considered, and an outline of the extension of the problem to higher dimensions given. Amusing alternative interpretations of the results are obtained. 
The apollonian octets and an inversive form of Krause's theorem
Abstract Suppose we are given three disjoint circles in the Euclidean plane with the property that none of them contains the other two. Then there are eight distinct circles tangent to the givenExpand
A case of the 3-dimensional problem of Apollonius
SummaryInn-dimensions the problem of Apollonius is to determine the (n−1)-spheres tangent ton+1 given (n−1)-spheres. In case no two of the given (n−1)-spheres intersect and no three have the propertyExpand
A new solution of Apollonius’ problem based on stereographic projections of Möbius and Laguerre planes
In this paper we give a new proof of Apollonius’ problem based on the stereographic projection in spherical model of Möbius geometry and cylinder model of Laguerre geometry.
N ov 2 01 9 IMPROVED BOUNDS FOR RESTRICTED PROJECTION FAMILIES VIA WEIGHTED FOURIER
  • 2019
It is shown that if A ⊆ R3 is a Borel set of Hausdorff dimension dimA > 1.5, then for a.e. θ ∈ [0, 2π) the projection πθ(A) of A onto the 2-dimensional plane orthogonal to 1 √ 2 (cos θ, sin θ, 1)Expand
Improved bounds for restricted projection families via weighted Fourier restriction.
It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A > 1.5$, then for a.e.~$\theta \in [0,2\pi)$ the projection $\pi_{\theta}(A)$ of $A$ onto the 2-dimensionalExpand
Apollonius by Inversion

References

Zum Problem des Apollonius