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.
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Lecture held at the Conference in memory of Cesarina Tibiletti Marchionna on November 7, 2006
Received: March 2007
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Scimemi, B. Central Points of the Complete Quadrangle. Milan j. math. 75, 333–356 (2007). https://doi.org/10.1007/s00032-007-0076-6
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DOI: https://doi.org/10.1007/s00032-007-0076-6