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Journal for Geometry and Graphics 22 (2018), No. 1, 001--013
Copyright Heldermann Verlag 2018

A Synthetic Way to Geometrize the Method of Coordinates

Giuseppina Anatriello
Dept. of Architecture, University of Naples, Via Monteoliveto 3, 80134 Naples, Italy

Horst Martini
Faculty of Mathematics, Technical University, 09107 Chemnitz, Germany

Giovanni Vincenzi
Dept. of Mathematics, University of Salerno, Via Giovanni Paolo 132, 84084 Fisciano, Italy

We propose a synthetic way (ensuing from Euclid's Elements) to geometrize the method of coordinates and thus to reformulate analytic geometry using a synthetic, axiomatic approach. In the theory that we will develop the segment arithmetic (Streckenrechnung) introduced by David Hilbert in his Grundlagen der Geometrie plays a crucial role. Analytic geometry has fundamental scientific and mathematical significance since, e.g., it is essential for the application of mathematics to physical and natural sciences. Our synthetic approach is certainly useful for a theoretical understanding of hierarchical structures of axiomatic theories, it can stimulate problem solving in the spirit of undergraduate mathematics, and it can even help to enhance classroom learning, all this being very important in modern times.

Keywords: Coordinate system, Euclidean geometry, analytic geometry, synthetic geometry, segment arithmetic, Theorem of Desargues, Theorem of Pappus.

MSC: 51M05; 51N20, 03A05

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