Central Points of the Complete Quadrangle |
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Authors: | Benedetto Scimemi |
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Affiliation: | (1) Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 32121 Padova, Italy |
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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. Received: March 2007 |
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Keywords: | KeywordHeading" >. Quadrangle central point symmetric function |
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