[pt] Esta Tese investiga mudanças ocorridas no texto de demonstração em geometria plana elementar, em livros-textos usados no ensino brasileiro a partir do século XIX. O trabalho está organizado em duas partes: a primeira, um estudo histórico de releituras dos Elementos de Euclides que, a partir do século XVI, em cada época influenciaram a constituição da matemática escolar em seu sentido mais amplo e a segunda, as análises das demonstrações. A base documental inclui obras históricas e livros relevantes no ensino-aprendizagem da geometria dedutiva no Brasil. As releituras de Euclides e os livros-texto permitiram estabelecer os parâmetros de análise das demonstrações e de inferências de ordem mais geral. As análises das demonstrações mostram evidências que confirmam mudanças de várias ordens: a redação, o método e a justificativa da prova. Foi possível identificar dois tipos básicos de livros-texto que denominamos elementos de geometria e livros de matemática, e caracterizar a correlação existente entre eles: o estudo dedutivo em geometria, originário no livro tipo elementos de geometria que se estrutura conforme o padrão euclidiano teorema- problema, se altera devido à interação geometria-álgebra- aritmética, a qual caracteriza o livro de matemática desde a sua origem até a atualidade. Foi possível constatar, ainda, que o estudo dedutivo em geometria tende a desaparecer quase totalmente nos textos tipo livro de matemática. O estudo - texto demonstrativo e suas mudanças - mostra uma história dos conteúdos matemáticos, dos livros-texto e da matemática escolar no Brasil. / [en] This thesis studies changes in proofs of elementary geometry theorems found in brazilian textbooks from the XVIIIth century on. The thesis is composed of two parts. The first one contains a historic survey of re-readings of Euclid´s Elements which, from the XVIth century on influenced the constitution of the school mathematics corpus, in its wide sense. The second ´part contains the actual analysis of proofs. We examine both historically important Works and books that were influential in the teaching and learning of deductive geometry in Brazil. The study of Euclid´s Elements and later works enabled us to set up the guidelines for the proofs analysis and all of more general inferences. The proofs analysis shows several types of change: the style, the method and the actual proof. It was possible to identify two basic kinds of text-books, which we call elements of geometry and mathematics books, respectively, and to establish the correlation between the two types: the deductive study of geometry, present in the elements of geometry, that adopted the Euclidean pattern of theorem - problem, changes due to the interaction between geometry, arithmetic and algebra, which characterizes the mathematics book, from their beginnings to the present. It was also possible to establish that the deductive study of geometry almost disappears in the mathematics books. The study of proofs and their changes shows the history of the contents of school mathematics, their text-books and of school mathematics in Brazil.

O estudo de impressos direcionados para professores, publicados pelos órgãos oficiais de Educação, contendo sugestões sobre os modos de fazer em sala de aula, constituindo uma literatura cinzenta escolar, aparecem como um instrumento eficaz para o aprofundamento dos estudos da História da Educação Matemática no Brasil e das relações entre programas, conteúdos e práticas escolares. Diante disso, o objetivo da tese foi problematizar de que modo foram construídas as propostas de alterações metodológicas para o ensino do número nas séries iniciais do Ensino Fundamental, no período entre 1961 e 1979, de modo a tentar compreender como foram produzidas as representações de ensino moderno, fundamentadas no ideário do Movimento da Matemática Moderna (MMM), nas publicações das Secretarias de Educação de São Paulo. E também, os modos de produção desses modelos, ou seja, a transformação na representação didático-pedagógica do conceito de número, no período analisado, nas orientações publicadas. Ainda, mais especificamente, busca-se entender como ocorre a apropriação dos estudos de Zoltan Paul Dienes nesses impressos. Acredito que o estudo que explore o diálogo entre passado e presente, que procura compreender as condições que permitiram a produção das representações sobre como ensinar e aprender Matemática pode subsidiar as problematizações diárias sobre a prática e possíveis novas propostas. Isso implica seguir e procurar desvendar os processos de apropriação utilizados pelos elaboradores das publicações, além de procurar caracterizar e diferenciar o MMM no ensino primário. O período histórico da pesquisa foi determinado pelas fontes selecionadas, após um levantamento das publicações existentes (Implantação da escola municipal de oito anos, de 1969, e os quatro volumes do Manual de Detalhamento de Currículo de 1974, 1976, 1977 e 1979, na memória técnica documental do Município de São Paulo e no Arquivo Pessoal Lucília Bechara Sanchez. A opção por essas fontes relaciona-se ao reconhecimento do valor atribuído às publicações no subsídio de professores para as mudanças, num período de expansão e criação dos sistemas de ensino no Brasil, com transformações na estrutura, no funcionamento, nos programas e no currículo de Matemática, de acordo com as normativas impostas pelas LDB 4.024/1961 e LDB 5.672/1971. Para complementar a análise, foi necessário problematizar as dificuldades do trabalho com essa literatura cinzenta escolar como fonte. A escassez de pesquisas que as utilizam pode ser explicada pela profusão desses textos, que, apesar de emanados de um mesmo órgão público, têm fases diferentes, consoante com os grupos que os produziram. Na articulação das questões, fiz uso da abordagem da história cultural e me apoiei nos conceitos de representação, apropriação e estratégias, postas por Chartier (1991) e Certeau (1982). Concluí que, no período estudado, as publicações produzidas pelas Secretarias foram utilizadas como estratégia de reformulação curricular e divulgação para implementar as novas diretivas para o ensino de Aritmética na escola primária paulista, adequando-se às recomendações dos novos campos da psicologia e da didática. A pesquisa ainda assinalou que a apropriação das ideias de Zoltan Dienes, defendendo uma abordagem estruturalista para a Matemática, produziu grandes reformulações na didática da Matemática, ressignificando o quê, como e pra quem ensinar. / The study of papers targeted to teachers, published by the official agencies of Education, containing suggestions on how to do in the classroom, making up a blurry school literature, appear as an effective instrument for deepening the studies of Mathematics Education History in Brazil and the relationships between programs, contents and school practices. Given the importance of these papers, this thesis aims to discuss how the proposals of methodological changes to the teaching of numbers in the early grades of elementary school were built. I want to understand how the representations were made of \"modern education\" based on the ideals of MMM, in the publications of the Departments of Education of Sao Paulo and the ways of production of these models. What transformation does the didactic teaching go through on the concept of number in the analyzed period through the guidelines published to the teachers by the Education Department? And more specifically, what reveals the papers of the Education Departments in terms of ownership of the studies conducted by Zoltan Paul Dienes? I believe that the questioning about what it was like to teach arithmetic and the methods suggested for this teaching in the initial grades may contribute to the understanding of the process of learning mathematics and how this process influenced - and continues to influence - the teaching of mathematics in the current educational context. The historical period of the research was determined by selected sources, after a survey of the existing publications (Implementation of the eight year municipal school, of 1969, and the four volumes of the MDC, of 1974, 1976, 1977 and 1979) in the Document Technical Memory of São Paulo and APLB. The choice of these sources is related to the recognition of the value assigned to the publications in the supplies of teachers for changes in a period of expansion and creation of educational systems in Brazil, with changes in the structure, operations, programs and in the Mathematics curriculum in accordance with the regulations imposed by BDL 4024/61 and BDL 5672/71. The work also included the comparison between publications with LDB/61 and the LDB/71, considering the place where the production of the official papers was held and the everyday of the production process, their dynamics and backstage. In order to complement the analysis, it was necessary to discuss the difficulties of working with this blurry school literature as a source. In articulating the issues, I use the approach of cultural history and lean against the concepts of representation, ownership and strategies put by Chartier (1991) and Certeau (1982). I conclude that during the studied period, the publications produced by the Departments of Education were used as a strategy adapting to the recommendations of the new fields of psychology and didactics. The survey also noted that ownership of the ideas of Zoltan Dienes, advocating a structuralist approach to mathematics, produced major reformulations in mathematics didactics, giving new meaning to teaching, how to teach and for whom to teach.

Submitted by Mario BC (mario@bc.ufrpe.br) on 2017-02-02T13:49:54Z No. of bitstreams: 1 Francisca Alves de Souza.pdf: 1429924 bytes, checksum: 8dcb63580a14736ed7ee893569075561 (MD5) / Made available in DSpace on 2017-02-02T13:49:54Z (GMT). No. of bitstreams: 1 Francisca Alves de Souza.pdf: 1429924 bytes, checksum: 8dcb63580a14736ed7ee893569075561 (MD5) Previous issue date: 2016-05-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Polynomials are of paramount importance to mathematics and when associated with functions modeling various phenomena of our day to day. This issue is addressed for the rst time at 8 years of elementary school, but students come to high school and college with various dificulties in learning it. These difi culties occur for many reasons, the main one being the aversion that students have the math classes and for that reason this work has as one of its objectives the use of the history of mathematics as a teaching support for teaching and polynomials learning. The other goal is to develop an educational support material for use in the classroom. To verify the e ectiveness of the use of the history of mathematics as a teaching resource in polynomials teaching a survey was conducted with students of the 1st series of technical course in informatics for integrated internet to high school the IFCE-Campus Crato, applying the following methodology: Application a test, which was called test 1, to check students' di culties in resolving issues surrounding the concept of polynomials, then held a workshop for polynomials study using the history of mathematics as a didactic support and to nish we applied a second test (test 2) to see if there was some progress in student learning. / Os polinômios são de suma relevância para a matemática e quando associados a funções modelam vários fenômenos do nosso dia a dia. Esse assunto é abordado, pela primeira vez, no 8º ano do ensino fundamental, porém os alunos chegam ao ensino médio e superior com várias dificuldades na aprendizagem do mesmo. Essas dificuldades ocorrem por inúmeros motivos, sendo um dos principais a aversão que os alunos tem pelas aulas de matemática e por essa razão este trabalho tem como um dos seus objetivos a utilização da história da matemática como suporte didático para o ensino e aprendizagem de polinômios. O outro objetivo é elaborar um material de apoio didático para ser utilizado na sala de aula. Para veri ficar a eficácia da utilização da história da matemática como recurso didático no ensino de polinômios foi realizada uma pesquisa com alunos da 1ª série do curso técnico em informática para internet integrado ao ensino médio do IFCE-Campus Crato, aplicando a seguinte metodologia: aplicação de um teste, o qual foi chamado de Teste 1, para verificar as dificuldades dos alunos na resolução de questões envolvendo o conceito de polinômios, depois foi realizada uma o cina para estudo de polinômios usando a história da matemática como suporte didático e para finalizar foi aplicado um segundo teste (Teste 2) para verificar se houve algum avanço na aprendizagem dos alunos.

O presente trabalho descreve uma proposta de aprendizagem significativa e contextualizada social e culturalmente na introdução do ensino de Álgebra por meio da utilização do filme Jogo da Imitação (2014), que traz elementos da História da Matemática, em especial da História da Criptografia. A proposta descrita apresenta o cinema como organizador prévio para as aulas de Matemática, em particular de Álgebra, com a finalidade de promover aprendizagem significativa, no sentido de Ausubel, Novak e Hanesian (1980). O trabalho também traz a descrição de atividades nas quais há um uso intencional da História da Matemática na construção do conhecimento para a formação do cidadão criativo, crítico, responsável e participativo, envolvendo professores de matemática e alunos do Ensino Fundamental II de uma escola particular da cidade de São Paulo. O trabalho também descreve a concepção de ensino dessa escola, em que as atividades foram realizadas, a fim de dar a elas o devido contexto. / The present work describes a meaningful, socially and culturally contextualized learning approach in the introduction of Algebra teaching through the use of the movie The Imitation Game (2014), which brings elements of the History of Mathematics, especially the History of Cryptography. The proposal described presents the cinema as a previous organizer for Mathematics classes, in particular Algebra, with the purpose of promoting meaningful learning, in the sense of Ausubel, Novak and Hanesian (1980). The paper also describes activities in which there is an intentional use of the History of Mathematics in the construction of knowledge for the formation of a creative, critical, responsible and participative citizen, involving mathematics teachers and Elementary School students from a private school in Sao Paulo City. The paper also describes the conception of teaching of this school, in which the activities were carried out, in order to give them the due context.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-06-16T14:36:33Z No. of bitstreams: 1 silviadecastrodebarros.pdf: 3863081 bytes, checksum: ab4a7e88d1525ed1bb94676937c74b2c (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T14:16:49Z (GMT) No. of bitstreams: 1 silviadecastrodebarros.pdf: 3863081 bytes, checksum: ab4a7e88d1525ed1bb94676937c74b2c (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T14:17:10Z (GMT) No. of bitstreams: 1 silviadecastrodebarros.pdf: 3863081 bytes, checksum: ab4a7e88d1525ed1bb94676937c74b2c (MD5) / Made available in DSpace on 2016-07-13T14:17:10Z (GMT). No. of bitstreams: 1 silviadecastrodebarros.pdf: 3863081 bytes, checksum: ab4a7e88d1525ed1bb94676937c74b2c (MD5) Previous issue date: 2015-11-10 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente trabalho investiga historicamente o ensino de Geometria na formação de professores primários em Minas Gerais, entre as décadas de 1890 e 1940. Procurou-se responder às questões: Quais conteúdos de Geometria eram ensinados aos futuros professores nas Escolas Normais? Como esse saber foi tratado na formação dos normalistas? A quais finalidades respondia a Geometria presente na formação? Para tanto, a História Cultural foi utilizada como ferramental teórico-metodológico, juntamente com aportes da História da educação matemática. As fontes analisadas foram: legislações para as Escolas Normais; cadernos de alunas da professora mineira Alda Lodi, uma das fundadoras da Escola de Aperfeiçoamento de Belo Horizonte; dois livros que abordam a Geometria e que constavam da biblioteca da referida professora; e exemplares da Revista de Ensino de Minas Gerais. No período abrangido por essa pesquisa fervilhava o movimento da Escola Nova, que convivia ainda com o Método Intuitivo materializado nas Lições de Coisas. A pesquisa evidenciou a presença reduzida da geometria plana e espacial na formação de normalistas, sobretudo quando comparada à Aritmética ou ao Desenho. Observou-se também o afastamento da Geometria da prática docente; sendo uma disciplina mais próxima da cultura do secundário, que da cultura profissional, servindo como aplicação para a Aritmética. / This paper historically investigates Geometry teaching of elementary school teachers’ training in Minas Gerais, Brazil, between the decades of 1890 from 1940. It tried to answer some questions: What were the geometry contents taught to those teachers in Normal Schools? How did they deal with this knowledge in elementary school teachers’ training? What were the purposes of geometry at that time? Therefore, it was used cultural history as theoretical and methodological tool, together with resources of the history of mathematics education. The analyzed sources were: Normal Schools legislation; female students’ notebooks from teacher Alda Lodi, one of the founders of Belo Horizonte Teaching Improvement School; two books that mentioned Geometry and were in that teacher’s library; and copies of the Minas Gerais Journal of Education. In the period comprised in this research the Free School Movement boomed, and it also coexisted with the Intuitive Method implemented in Primary Object Lessons. The research showed the poor presence of flat and spatial Geometry in elementary school teachers’ training, especially when compared to Arithmetic or Drawing. We also observed a distance between Geometry and teaching practice, which became a subject of high school culture rather than of professional culture, serving as application for the arithmetic.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-07-05T11:54:32Z No. of bitstreams: 1 mariadasgracasschinnigerassungarcia.pdf: 1134387 bytes, checksum: 64c073ab58c8c7633340840516ced970 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-07-06T14:14:47Z (GMT) No. of bitstreams: 1 mariadasgracasschinnigerassungarcia.pdf: 1134387 bytes, checksum: 64c073ab58c8c7633340840516ced970 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-07-06T14:15:06Z (GMT) No. of bitstreams: 1 mariadasgracasschinnigerassungarcia.pdf: 1134387 bytes, checksum: 64c073ab58c8c7633340840516ced970 (MD5) / Made available in DSpace on 2018-07-06T14:15:06Z (GMT). No. of bitstreams: 1 mariadasgracasschinnigerassungarcia.pdf: 1134387 bytes, checksum: 64c073ab58c8c7633340840516ced970 (MD5) Previous issue date: 2018-04-02 / Esta dissertação estuda historicamente a presença do Desenho como matéria escolar no curso primário. Tomou-se como questão norteadora do estudo: Quais as finalidades da matéria Desenho lidas nas diretivas oficiais nacionais e mineiras das décadas de 1940-1950? No recorte temporal adotado – as décadas de 1940 e 1950 – o ensino é regido pela Lei Orgânica do Ensino Primário, de 1946. O estudo se situa no campo da História da educação matemática, tomando como referenciais teóricometodológicos aportes advindos da História Cultural, e considerando-se a cultura escolar e as disciplinas ou matérias como objetos históricos. Foram examinados, além da Lei Orgânica para o Ensino Primário, os Programas em Experiência de Minas Gerais, publicados na década de 1940 e republicados até 1961. Esses Programas trouxeram o Desenho integrado às áreas de ensino, apresentado como matéria auxiliar, que aparece como atividade de expressão, observação e intuição. Assume características rudimentares, ou seja, passa a constituir um ensino consubstanciado com o caráter de iniciação aos saberes escolares; sem nada a dever aos saberes de referência. Pautado nas finalidades dadas para o ensino primário, se mostrou alinhado às matérias escolares constitutivas deste ciclo escolar – aritmética, geometria, língua pátria, história e geografia, ciências e higiene. / This dissertation historically studies the presence of Design as primary school subject matter. It was taken as the guiding question of the study: what are the finities of the drawing matter read in the official national and mining directives of the 1940-1950 decades? In the 1940s and 1950s, the teaching is governed by the Organic Law of Primary Education of 1946. The study is situated in the field of History of Mathematics Education, taking as theoretical-methodological references contributions from Cultural History, and considering school culture and disciplines or subjects as historical objects. In addition to the Organic Law for Primary Education, the Programs in Experience of Minas Gerais, published in the 1940s and republished until 1961, were examined. These programs brought the Integrated Design to teaching areas, presented as an auxiliary material, which appears as an activity of expression, observation and intuition. It assumes characteristics of a discipline based on the perspective of rudiments. By these considerations, it begins to constitute a teaching consubstantiated with the character of initiation to the school knowledge; with nothing to do with reference knowledge. Targeted at the primary education objectives, it was aligned with the school subjects constituting this school year - arithmetic, geometry, mother tongue, history and geography, science and hygiene

When the effects of wind and air pressure combine with a high tide to give unusually high water levels this can lead to severe coastal flooding. This happened in England in early 1953 when 307 people died in the East Coast Flood. In Britain today such events, now called storm surges, are forecast daily using computer models from the National Oceanographic Centre in Liverpool, formerly the Liverpool Tidal Institute (TI). In 1919, when TI was established, such events were considered unpredictable. TI's researchers, Joseph Proudman (1888-1975), Arthur Doodson (1890-1968), Robert Henry Corkan (1906-1952) and Jack Rossiter (1919-1972), did much mathematical work to attempt to change this. In 1959 Rossiter published a set of statistical formulae to forecast storm surges on the East Coast and a national warning system was predicting such events using these formulae. At this point TI believed they had made surges at least as predictable as they could with their existing methods. This thesis provides a narrative of how this perceived rise in the predictability of surges happened, analysing how TI worked to achieve it between 1919 and 1959 by following two interwoven, contingent and contested threads: practices of calculation and patronage. A key aspect of this thesis is the attention I pay to material practices of calculation: the methods, technologies and management practices TI's researchers used in their mathematical work on storm surge forecasting. This is the first study by historians of oceanography or meteorology that pays this detailed level of attention to such practices in the construction of forecasting formulae. As well as using published accounts, I analyse statistical research in the making, through notes, calculations, graphs and tables produced by TI's researchers. They used particular practices of calculation to construct storm surges as calculable and predictable scientific objects of a specific kind. First they defined storm surges as the residuals derived from subtracting tidal predictions from observations. They then decided to use multiple regression, correlating their residuals with pressure gradients, to make surges predictable. By considering TI's practices of calculation the thesis adds to the literature on mathematical research as embodied and material, showing how particular practices were used to make a specific phenomenon predictable. I combine this attention to mathematical practice with analysis of why TI's researchers did this work. US historians have emphasised naval patronage of physical oceanography in this period but there is very little secondary literature for the British case. The thesis provides a British case study of patronage of physical oceanography, emphasising the influence on TI's work not only of naval patronage but also of local government, civil state and industrial patronage. Before TI's establishment Proudman argued that it should research storm surges to improve the Laplacian theory of tides. However, when the new Institute received patronage from the local shipping industry this changed and the work on forecasting surges was initially done as part of a project to improve the accuracy of tidal predictions, earning TI further patronage from the local shipping industry. After a flooding event in 1928 the reasons for the work and the patronage again shifted. Between then and 1959 TI did this work on commission from various patrons, including local government, civil state and military actors, which connected their patronage to national debates about state involvement in flood defence. To understand why TI's researchers worked on forecasting surges I analyse this complex mix of patrons and motivations. I argue that such complex patronage patterns could be fruitfully explored by other historians to further existing debates on the patronage of oceanography.

Students today need to be taught not only the real life context of their mathematics lessons but also the historical context of the theory behind their mathematics lessons. Using history to teach mathematics, makes your lessons not only interesting but more meaningful to a large percentage of your students as they are interested in knowing the who, how and why about certain rules, theorems, formulas that they use everyday in class. Students are captivated by learning the history behind mathematicians, rules, etc. and therefore can link the lesson to something in history and a concept. Even learning the mathematics behind historical events motivates and interests them. They cannot get enough!

This diploma thesis deals with the comparison of historical solutions of word problems with student solutions. Its aim was to describe how students solve historical word problems, while looking for analogies between student and historical solutions. This intention led me to a better understanding of student solutions. The theoretical part of the thesis describes important concepts for algebraic word problems, such as a variable, algebraic expression or algebraic word problem. In the historical part I describe chronologically the development of algebra from antiquity through the Middle Ages and the Renaissance to the Baroque. In each period, I mention important mathematicians of the time and present several solved word problems. In most cases, I analyze these solutions from the perspective of today's mathematics. The theoretical part describes the research that took place at the eight-year grammar school. As part of the research, I gave students 6 historical tasks across historical periods and then analyzed the ways in which students solved problems. I found that for most of the tasks, there were solutions similar to the historical solutions among the student's solutions. Some historical methods appeared very often. An example is the use of addition instead of multiplication, or division, as used by the...

[pt] Este trabalho tem como objetivo principal apresentar e demonstrar os teoremas de Miquel que tratam de retas, círculos e suas interseções, assim como a versão de Clifford para os mesmos. Mais especificamente do teorema referente ao pentágono que afirma que dado um pentágono, o prolongamento dos seus lados formam cinco triângulos e os círculos circunscritos a esses triângulos se intersectam dois a dois e os pontos de interseção distintos dos vértices estão sobre uma mesma circunferência. Os teoremas de Miquel são demonstrados de forma original, com exceção do teorema citado, cuja prova é igual àquela do artigo original, a menos de mudanças de notação e maior detalhamento de argumentos. A versão de Clifford para esse teorema é provada apenas com o uso de argumentos de geometria euclidiana, diferente do proposto em seu artigo, que lança mão de ferramentas da geometria projetiva e das curvas algébricas para chegar à sua tese. Também é feita uma demonstração para a generalização do teorema acima ao se tomar n retas. Além disso, este trabalho apresenta uma proposta de atividades pedagógicas com o uso do software de geometria dinâmica GeoGebra, como ferramenta facilitadora à visualização e dedução dos teoremas mais importantes do trabalho. / [en] This work aims to present and demonstrate Miquel s theorems dealing with straigt lines, circles and their intersections, as well as Clifford s version of the same theorems. More specifically regarding the theorem that makes reference to the pentagon, which asserts that given a pentagon, the extension of its sides form five triangles and the circles circumscribed to these triangles intersect two by two, and the intersection points, not considering the vertices, are on the same circumference. Miquel s theorems are presented in an original way, with the exception of the above theorem, which is equal to the original one, apart from little changes of notation and more detailed arguments. Clifford s version of this theorem is presented with the use of Euclidean geometry arguments differing from the one proposed in his article, which makes use of tools of projective geometry and algebraic curves to get to his thesis. There is also a demonstration for the generalization of the above theorem when n straigt lines are taken. In addition, this work proposes a pedagogical activity using the dynamic geometry software GeoGebra, as a facilitating tool for viewing and deduction of the most important theorems presented in this work.