Máximos e Mínimos na Geometria Euclidiana

Abstract

In this study we address classical problems of maxima and minima in Euclidean geometry involving areas and perimeters of triangles and polygons. First, we study some preliminary results and present their demonstrations using only basic concepts of Euclidean geometry. Among the problems studied here, we highlight the Isoperimetric Inequality for polygons and the general Isoperimetric Problem in the plane, historically associated with the legend of Princess Dido and the founding of Carthage. The maximum and minimum theory of Differential Calculus is a powerful tool for solving the problems presented. However, whenever possible, it has been replaced by features of elementary geometry, in order to make reading comprehensible even for high school students. We will also prove, in the appendix of this work, a particular property of the ellipse taking advantage of results shown previously. The work as a whole is targeted at elementary and middle school teachers who can select one or another topic to motivate their students and enrich their classes