O problema de Neusis e os elementos de Euclides: uma proposta de investigação para o ensino da geometria

Abstract

The frustrated attempts to solve the problem of the trisection of any angle resulted in innumerable mathematical discoveries and served as a source for the emergence of new ideas and unknown paths. It is in this universe of trial and error that this research is found. Even knowing that Wantzel had proven it to be a mathematically impossible task to perform, we will study the challenge of trisecting (finding the third part) any angle using only a ruler and compass, through the neusis problem, as defined by the Greeks, by applying the theoretical basis of the postulates described in the first book (or chapter) of Euclid’s most famous work: “The elements”. For this purpose, we made a theoretical overview of the propositions of Euclid's book I, the history of trisection, as well as the already known applications for the neusis problem. By using these frameworks as a theoretical background, in the trial-and-error method, we have found not only a new way to make some exact trisections but satisfactory approximations for acute angles as well, always obeying the construction rules of the Euclidean elements. The theoretical overviews, the exact trisections and the approximations were organized for presentation following the script of a non-rigid didactic sequence, aiming to be applied as a geometry teaching proposal for elementary, high school or college. At the end of this paper, we present some exercises of exact trisections and some approximations. Appendix B shows a didactic sequence structure with the theme of this research to assist the teacher who wishes to use it as a support for teaching geometry.