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Milan Journal of Mathematics

, Volume 75, Issue 1, pp 333–356 | Cite as

Central Points of the Complete Quadrangle

  • Benedetto ScimemiEmail author
Article
  • 49 Downloads

Abstract.

Generalizing the classical geometry of the triangle in the Euclidean plane E, we define a central point of an n-gon as a symmetric function E n E which commutes with all similarities. We first review various geometrical characterizations of some well-known central points of the quadrangle (n = 4) and show how a look at their mutual positions produces a morphologic classification (cyclic, trapezoidal, orthogonal etc.). From a basis of four central points, full information on the quadrangle can be retrieved. This generalizes a problem first faced by Euler for the triangle. Reconstructing a quadrangle from its central points is a geometric analogue of solving an algebraic equation of degree 4: here the diagonal triangle plays the role of a Lagrange resolvent and the determination of loci for the central points replaces the examination of discriminants for real roots.

Keywords.

Quadrangle central point symmetric function 

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Copyright information

© Birkhaueser 2007

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità di PadovaPadovaItaly

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