A reta de Nagel : uma abordagem geométrica e algébrica de um alinhamento notável

Abstract

This work aims to address important concepts of Euclidian Geometry originally from the study of the triangle and its remarkable points, giving shape to our main object which is the Nagel Point and the Spieker Circle. By that, we seek to stimulate maths teachers and students, particularly in high school, to work in the classroom with this geometry, aiming to improve the processes of teaching and learning of mathematics. Initially, we present necessary introductory concepts to our study, using Flat Euclidean Geometry. Then, we show the results that characterize the remarkable points of the triangle through the properties and concepts previously seen that are necessary to the construction of our object of study. We show a comparative picture between the Euler and Nagel straight lines, highlighting their main properties and characteristics. In order to explore new concepts, we extend our study by taking Euler’s Line and the Circumference of Nine Points of Tores Rojas and Mendoza. We approach Affine Geometry, presenting essential basic definitions, forming a necessary language to speak about Linear Algebra focusing on geometric concepts that are developed and used. We show that the universal property of points in the plane is fundamental for characterizing the remarkable points of a triangle. We use algebra and vector methods to obtain the statements and results related to our study object.