O problema das quatro guaritas: Uma oportunidade para o estudo dos quadriláteros bicêntricos

Abstract

This work uses an original problem situation, entirely elaborated by the author, called "The problem of the four security cabins", and, from the discussion of its solution, intends to study the bicentric quadrilaterals, figures of great aesthetic beauty and that, due to the peculiarities characterized by the fact that they are both cyclical and tangential, make it possible to recall some very important subjects in the teaching - learning context of Euclidean Geometry, such as the Similarity of Triangles, the Loci, the Internal Bisector Theorem, Pitot's theorem, the Angles in a Circle, the Theorems of Ptolemy and Hipparchus, and a few others. We offer the reader an historical perspective of the teaching of Euclidian Geometry in Brazil, including what refers to the content of textbooks, especially of the past and present centuries. We use tools such as Manual Geometric Drawing (ruler and compass), and digital devices, more specifically free dynamic geometry software Geogebra, to analyze the bicentric quadrilaterals, and also include a series of activities that can eventually be adopted by Mathematics teachers in their classes of the Eighth and Nineth Years of Primary Education