Taherian, S. Gh.; Mohseni Takaloo, S. A new solution of Apollonius’ problem based on stereographic projections of Möbius and Laguerre planes. (English) Zbl 1421.51001 Beitr. Algebra Geom. 60, No. 3, 465-469 (2019). The Apollonius problem of finding circles tangent to three given circles in the Euclidean plane, is a well-known question for which different proofs exist. The authors present a new solution based on stereographic projection using the spherical model of Möbius geometry and the cylinder model of Laguerre geometry. Reviewer: Dirk Keppens (Gent) Cited in 1 Document MSC: 51B10 Möbius geometries 51H15 Topological nonlinear incidence structures 51B15 Laguerre geometries Keywords:Möbius planes; Laguerre planes; Apollonius’ problem; stereographic projection PDFBibTeX XMLCite \textit{S. Gh. Taherian} and \textit{S. Mohseni Takaloo}, Beitr. Algebra Geom. 60, No. 3, 465--469 (2019; Zbl 1421.51001) Full Text: DOI References: [1] Benz, W.: Vorlesungen über Geometrie der Algebren, Band 197. Springer, Berlin (1973) · Zbl 0258.50024 [2] Blaschke, W.: Analytische Geometrie. Verlag Birkhäuser, Basel (1954) · Zbl 0057.12705 [3] Bruen, A., Fisher, J.C., Wilker, J.B.: Apollonius by inversion. Math. Mag. 56, 97-103 (1987) · Zbl 0525.51019 [4] Budai, L.: A possible general approach of the Apollonius problem with the help of GeoGebra. Ann. Math. Inform 40, 163-173 (2013) [5] Coxeter, H.S.M.: The problem of Apollonius. Am. Math. Mon. 58, 5-15 (1968) · Zbl 0161.17502 [6] Fitz- Gerald, J.M.: A note on a problem of Apollonius. J. Geom. 5, 15-26 (1974) · Zbl 0288.50006 [7] Heise, W.: Bericht über \[\kappa\] κ -affine Räume. J. Geom. 1, 197-224 (1971) · Zbl 0228.50033 [8] Hoshen, J.: The GPS equation and the problem of Apollonius. IEEE Trans. Aearospace Electron. Syst. 32, 1116-1124 (1996) [9] Knight, R.D.: The Apollonius contact problem and Lie geometry. J. Geom. 83, 137-157 (2005) · Zbl 1093.51006 [10] Lewis, R.H., Bridgell, S.: Conic tangency equation and Apollonius problem in biochemistry and pharmacology. Math. Comput. Simul. 61, 101-104 (2003) · Zbl 1016.51014 [11] Sasaki, Ch.: Descartes’s Mathematical Thought. Springer, New York (2003) · Zbl 1045.01001 [12] Schroth, A.E.: Topological Circle Planes and Topological Quadrangles, Pitman Research Notes in Mathematics Series, vol. 337. Longman, Harlow (1995) · Zbl 0839.51013 [13] Stoll, V.: Zum Problem des Apollonius. Math. Ann. 6, 613-632 (1873) · JFM 06.0326.01 [14] Taherian, S. G.: Koordinatisierung miquelscher Benz-Ebenen und ihre Anwendungen in der Kryptologie, Ph.D. thesis TUM (2001) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.