A synthetic way to geometrize the method of coordinates

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