Teorema de Borsuk no plano

Abstract

This paper deals with the Borsuk Theorem, focusing on dimension 2. The theorem revolves around the question: "What is the smallest number of parts that a region can be divided into, to ensure that in each part the diameter is less than the diameter of the initial region?" Borsuk proves that the required number of divisions in the plan is less than or equal to 3, to ensure smaller diameter regions. In this paper we present a proof for the theorem above. When creating the minicourse “Borsuk Theorem in the Plane” and applying it to senior students from Jenny Gomes State School, it was proposed to review fundamental concepts of students’ prior knowledge in Plane Geometry, determine core deficiencies in these concepts and make students eager to acquire investigation and commitment around the subject, besides handling a content yet-unseen by them, presenting also the historical scenario of the theorem. Students enrolled willingly in the course. For data collection it was used a socioeconomic questionnaire, motivational basis tests and knowledge tests, before and after the course. The Borsuk theorem in the plan makes use only of elementary geometry and can be understood by high school students. New concepts such as diameter of a flat figure, lines of support and Pall Lemma will be presented. It was found that the activity is an effective tool against the disinterest and difficulty of students regarding Geometry.