Construções geométricas e a trissecção de um ângulo

Abstract

Geometric constructions, throughout history, have played a crucial role in consolidating fundamental concepts of Euclidean geometry. However, little attention has been given to their application in education. In this context, geometric constructions using only an ungraduated ruler and compass present not only a challenge but also an unsolvable problem in certain cases. As the theoretical framework for this research, we draw on Boyer and Merzbach (1974), Barbosa (2013), Papa Neto (2017), among others. This study aims to explore some geometric constructions with an ungraduated ruler and compass, such as parallel lines, angle bisectors, and the trisection of a right angle. It is important to highlight that certain elements cannot be constructed using ruler and compass, most notably the duplication of the cube, the squaring of the circle, and the trisection of an angle. For the latter, a false method is presented. The results show that the problem of angle trisection in geometric constructions represents an important development within Euclidean geometry, reflecting the challenges of logical rigor and the validation of methods in scientific progress