Uma abordagem de problemas da geometria plana do ponto de vista da geometria analítica

Abstract

In general, in mathematics olympiads, the subjects collected are organized into four major groups: algebra, combinatorics, number theory, and flat geometry. This last group, in particular, is an inexhaustible source of interesting problems. Solving an Olympic problem of geometry is a task that requires a solid knowledge of the propositions and theorems related to it. In some cases, appropriate geometric constructions must be considered in order to optimize the search for a solution. Sometimes trigonometric resources can be employed for the same purpose. Still, even considering all the geometric apparatus available to the student, many problems seem to be insoluble, whereas the use of a certain technique is not always evident. In our work, we will study geometric problems extracted from mathematical olympiads around the world and analyze them through two different approaches. In the first place, we will exhibit purely Euclidean solutions, so to speak. On the other hand, we will present algebraic solutions, that is, on the basis of Cartesian geometry. In some cases, we will use the methods of Differential and Integral Calculus, given its close relationship with Descartes geometry.