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Interpretação geométrica dos problemas clássicos de Desargues, Fagnano e Malfatti
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Abstract(s)
A presente dissertação de mestrado expõe um estudo que aborda as propriedades geométricas relacionadas com os problemas de Desargues, Fagnano e Malfatti, no domínio de diferentes geometrias. Em termos estruturais, o trabalho apresenta-se dividido em sete capítulos. O primeiro capítulo é dedicado à introdução. No segundo capítulo são introduzidos os conceitos teóricos básicos e as principais ferramentas utilizadas referentes às geometrias não euclidianas envolvidas no estudo. Atendendo à natureza dos problemas selecionados, são abordadas a geometria projetiva, a geometria hiperbólica e a geometria inversiva. Nos três capítulos seguintes, é apresentado um estudo geométrico dos referidos problemas, onde se exploram as respetivas propriedades e eventual aplicação. Destacamos, sempre que possível, as características específicas que despertaram o interesse dos matemáticos, perpetuando o seu estudo até aos nossos dias. O terceiro capítulo é dedicado ao problema de Desargues, mais conhecido pelo teorema dos dois triângulos. No quarto capítulo estudamos o problema de Fagnano, que consiste em inscrever num triângulo acutângulo um outro triângulo que tenha o mínimo perímetro possível. No quinto capítulo exploramos o problema de Malfatti, também relacionado com o triângulo, mas cujo objetivo é o de inscrever num triângulo dado três círculos, sendo cada um tangente externamente aos outros dois e simultaneamente tangente a dois lados do triângulo. Em cada um desses capítulos, apresentamos as referências históricas relacionadas com os problemas e respetivos autores. O sexto capítulo é dedicado à apresentação do resultado prático desenvolvido com base no estudo efetuado, o Jogo do Paralelo. Este jogo combina a figura geométrica comum aos três problemas, o triângulo, com o conceito de paralelismo, que diferencia a geometria euclidiana das geometrias não euclidianas. No trabalho de pesquisa e investigação efetuado, presente de forma sintética e clara nos capítulos terceiro a quinto, foi utilizado, como recurso tecnológico, o programa informático Geogebra 4.2. No capítulo final, destacamos uma visão conjunta dos problemas estudados, bem como os possíveis contributos do presente trabalho no desempenho docente da autora.
ABSTRACT: This masters’ degree thesis presents a study that addresses the geometrical properties that are related with the Desargues, Fagnano e Malfatti problems, taking different point of views. Regarding the structure, this study is divided into seven chapters. The .first chapter is the introduction. On the second chapter the aim is to introduce the basic theoretical concepts and the main tools used in what concerns the non euclidian geometries that are the object of this study. Taking in consideration the problems that were pointed out, the projective geometry the hyperbolic geometry and the inversive geometry will be studied. On the next three chapters, it will be presented a geometrical study of the mentioned problems, where their properties and applications will be explored. The specific characteristics that have always been mathematicians’ concern throughout the years will also be emphasized. Chapter three is dedicated to Desargues problem, which is best known by the two triangle theorem. On chapter four it will be studied the Fagnano problem, which consists in inscribing in an acute triangle another triangle that has the minimal possible perimeter. On chapter five, the Malfatti problem will be explored. This problem is also related to the triangle, though the aim is to inscribe three circles in a given triangle, being each circle externally tangent to the other two, and simultaneously tangent to two sides of the triangle. In each of the mentioned chapters, historical references related to the problems and each of the authors will be pointed out. On chapter six it will be presented a study case based on the "Parallel Game". This game combines the geometrical figure, which is common to the three problems - the triangle, with the concept of parallelism, which distinguishes the euclidian geometry from the non euclidianones. The technological resource used on the research and investigation process, which is clearly noticed from chapters’ three to five, was the computer program Geogebra 4.2. On the final chapter, the combined vision of the problems that were studied will be highlighted, as well as the possible contributes of this study to the authors’ daily teaching.
ABSTRACT: This masters’ degree thesis presents a study that addresses the geometrical properties that are related with the Desargues, Fagnano e Malfatti problems, taking different point of views. Regarding the structure, this study is divided into seven chapters. The .first chapter is the introduction. On the second chapter the aim is to introduce the basic theoretical concepts and the main tools used in what concerns the non euclidian geometries that are the object of this study. Taking in consideration the problems that were pointed out, the projective geometry the hyperbolic geometry and the inversive geometry will be studied. On the next three chapters, it will be presented a geometrical study of the mentioned problems, where their properties and applications will be explored. The specific characteristics that have always been mathematicians’ concern throughout the years will also be emphasized. Chapter three is dedicated to Desargues problem, which is best known by the two triangle theorem. On chapter four it will be studied the Fagnano problem, which consists in inscribing in an acute triangle another triangle that has the minimal possible perimeter. On chapter five, the Malfatti problem will be explored. This problem is also related to the triangle, though the aim is to inscribe three circles in a given triangle, being each circle externally tangent to the other two, and simultaneously tangent to two sides of the triangle. In each of the mentioned chapters, historical references related to the problems and each of the authors will be pointed out. On chapter six it will be presented a study case based on the "Parallel Game". This game combines the geometrical figure, which is common to the three problems - the triangle, with the concept of parallelism, which distinguishes the euclidian geometry from the non euclidianones. The technological resource used on the research and investigation process, which is clearly noticed from chapters’ three to five, was the computer program Geogebra 4.2. On the final chapter, the combined vision of the problems that were studied will be highlighted, as well as the possible contributes of this study to the authors’ daily teaching.
Description
Dissertação de Mestrado, Matemática para Professores, 17 de Outubro de 2013, Universidade dos Açores.
Keywords
Girard Desargues (1593-1662) Giovanni Fagnano (1715-1797) Gian Francesco Malfatti (1731-1807) Geometria Matemática
Citation
Faria, Raquel Maria Almeida. "Interpretação geométrica dos problemas clássicos de Desargues, Fagnano e Malfatti". 2013. XIV, 116 p.. (Dissertação de Mestrado em Matemática para Professores) - Ponta Delgada: Universidade dos Açores, 2013.