From Pure Geomatics for Algebraic Procedures with a View to Obtaining Equations from Ellipse from the Perspective of René Descartes

This article aims to study the ellipse from the perspective of pure or synthetic geometry to the representation of points on a plane through the use of real numbers, as well as the representation and classification of this conic curve through the use of equations. The perspective developed in this article is based on the view of Rene Descartes, in considering that “the algebraic steps in a demonstration should really correspond to a geometric representation.” The relevance of this article is to bring a reflection that eliminates the study of Analytical Geometry through ready-made and finished formulas, without satisfactory justification and without a logical chain that gives a greater meaning to the studied concepts. In general, the study developed in this article emphasizes the demonstration of results based on propositions adapted a priori, whose ability to be developed is aimed at establishing an "if...then" type of reasoning, making conjectures involving various knowledge already acquired and confirming such truths from a logical system, using definitions and propositions. Therefore, the demonstrations made in the scope of Synthetic Geometry will help to establish a connection with the equations obtained from the perspective of Analytical Geometry, serving as a consultation for students and professors of Analytical Geometry, thus avoiding sudden transitions between contents of degrees of distinct difficulties.