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1 results

  • Plane polygons revisited

  • B. H. Neumann
  • Journal:
    Journal of Applied Probability / Volume 19 / Issue A / 1982
    Published online by Cambridge University Press:
    14 July 2016, pp. 113-122
    Print publication:
    1982

  • A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.