Abstract
As is well known [1, p.480], the cycles and spears of the real Laguerre plane can be represented by the points and null planes of three-dimensional Minkowski space. Miquel's theorem in the Laguerre plane can then be expressed as an intrinsically interesting Minkowski space theorem the octahedron theorem. We outline the correspondence between the two theorems, and then give a metric vector space proof of the octahedron theorem, thereby providing an alternate proof of Miquel's theorem. We then discuss the generalization of both theorems to more general spaces.
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References
KUNLE, H., and FLADT, K.,Erlanger program and higher geometry, inFundamentals of Mathematics II Geometry, ed. H. Behnke, F. Bachmann, K. Fladt, H. Kunle, The MIT Press, Cambridge, Mass. U.S.A 1983
SNAPPER, E., and TROYER, R. J.,Metric Affine Geometry, Academic Press, New York, 1971
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Lester, J.A. The octahedron theorem in Minkowski three-space a metric vector space proof of Miquel's theorem in the Laguerre plane. J Geom 30, 196–202 (1987). https://doi.org/10.1007/BF01227817
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DOI: https://doi.org/10.1007/BF01227817