Mémoire de Maîtrise
DOI
10.11606/D.55.2013.tde-18042013-141806
Document
Auteur
Nom complet
Elvis Donizeti Neves
Unité de l'USP
Domain de Connaissance
Date de Soutenance
Editeur
São Carlos, 2013
Directeur
Jury
Ribeiro, Hermano de Souza (Président)
Dias, Ires
Malagutti, Pedro Luiz Aparecido
Titre en portugais
Caracterização e localização dos pontos notáveis do triângulo
Mots-clés en portugais
Números complexos
Pontos notáveis do triângulo
Sistema cartesiano
Triângulos
Resumé en portugais
Titre en anglais
Characterization and location of the notable points of the triangle
Mots-clés en anglais
Cartesian system
Complex numbers
Notable points of the triangle
Triangles
Resumé en anglais
The teaching of Mathematics is generally guided by the procedures contained in the textbooks. Thus, the organization of the mathematical concepts in these books should be able to allow the reader to interpret the Mathematics in its essence, admitting the establishment of relationships between the contents. However, what is observed in the materials is a conglomeration of disparate definitions and concepts that lead the reader to learning difficulties in the area. For this reason, this work aimed to locate and characterize the notable points of the triangle: the centroid or barycenter (G), the orthocenter (H), the circumcenter (O), the center (N) of circumference of nine points, three former centers of the ex-inscribed circles, orthogonal projections of the vertices on the opposite sides and the points of tangency of the inscribed and the ex-inscribed circumference. Four approaches are presented to achieve these goals: a-) to introduce the geometry of the triangle using visual perception techniques, b-) to characterize some notable points of the triangle, as points of maximum or minimum of functions with the demonstrations using the Cauchy-Schwarz inequality and between the arithmetic and geometric mean;-c) to use a suitable Cartesian system for calculating the abscissas and ordinates of the centroid (G), of orthocenter (H) and of the circumcenter (O) of a triangle;-d) to use complex numbers for the complete location of all notable points of the triangle, beyond depicting the Euler equation of the line, the incenter (I) and the three former centers IA, IB and IC located in simple formulas. The work is concluded with the Feuerbach's Theorem, presented with an elementary proof, showing that the nine-point circle and the incircle is tangent internally and that the circumference of the nine points is externally tangent to each of the three ex-inscribed circles and the Napoleons Theorem, in which the barycenters of equilateral triangles, constructed from the sides of any triangle, form another equilateral triangle. Comparing the approaches detached hitherto, the conclusion is that the understanding of complex numbers paradoxically simplifies troubleshooting of plane geometry and the solution of polynomial equations. Thus, it is believed that further exploration of this content in mathematics education could make learning more attractive and simplified

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elvisresumida.pdf (2.11 Mbytes)
Date de Publication
2013-04-18

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