Geodesic
A Synthetic Way to Geometrize the Method of Coordinates
Journal for geometry and graphics, Tome 22 (2018) no. 1, pp. 1-13 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 51M05, 51N20, 03A05
Mots-clés : Coordinate system, Euclidean geometry, analytic geometry, synthetic geometry, segment arithmetic, Theorem of Desargues, Theorem of Pappus 
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G. Anatriello; H. Martini; G. Vincenzi . A Synthetic Way to Geometrize the Method of Coordinates. Journal for geometry and graphics, Tome 22 (2018) no. 1, pp. 1-13. http://geodesic.mathdoc.fr/item/JGG_2018_22_1_JGG_2018_22_1_a0/