Pontos Notáveis do triângulo
Neste livro, exploraremos os principais pontos notáveis do triângulo: incentro, circuncentro, ortocentro e baricentro. Apresentaremos construções, propriedades, demonstrações e exercícios.
Table of Contents
- Incentro
- Incentro
- Circuncentro
- Circuncentro
- Ortocentro
- Ortocentro
- Baricentro
- Baricentro
- Exercícios sobre os pontos notáveis
- A reta de Euler e o centro de massa do triângulo
Incentro
Incentro
O incentro de um triângulo é o ponto de encontro das bissetrizes dos ângulos internos do triângulo.
Construção do Incentro do Triângulo
[list][*]Ative a ferramenta POLÍGONO (Janela 5) e clique em três lugares distintos para formar um triângulo. Para fechar o triângulo clique novamente no primeiro ponto. Naturalmente que os pontos não podem estar alinhados. Um triângulo com vértices nos pontos A, B e C será criado.[/*][*]Ative a ferramenta BISSETRIZ (Janela 4) e clique sobre os vértices: A, C e B (nessa ordem). Posteriormente sobre os vértices C, B e A (nessa ordem). Duas bissetrizes foram criadas com os nomes “d” e “e”.[/*][*]Ative a ferramenta INTERSEÇÃO DE DOIS OBJETOS (Janela 2) e crie o ponto D de interseção das retas “d” e “e”. [br][/*][*]Queremos traçar a terceira bissetriz. A pergunta é: será que ela também passará pelo ponto D? Ative a ferramenta BISSETRIZ (Janela 4) e clique nos pontos B, A e C (nessa ordem).[br][/*][/list]
Construção do círculo inscrito
[list][*]Ative a ferramenta EXIBIR/ESCONDER OBJETO (Janela 11), clique sobre as retas d, e, f e aperte ESC posteriormente. [br][/*][*]Vamos modificar o nome do ponto D para Incentro. Para tal, clique com o botão do lado direito do mouse sobre o ponto D e selecione a opção RENOMEAR. Na nova janela que aparecerá, escreva Incentro e clique em OK.[/*][*]Ative a ferramenta RETA PERPENDICULAR (Janela 4), clique no ponto Incentro e no lado c do triângulo (que liga os pontos A e B). [br][/*][*]Ative a ferramenta INTERSEÇÃO DE DOIS OBJETOS (Janela 2), clique na reta g e, posteriormente, no lado c que liga os pontos A e B. Um ponto D será criado[1]. [br][/*][*]Nosso interesse é apenas no pé da perpendicular (ponto D). Assim, podemos esconder a reta g, usando a ferramenta EXIBIR/ESCONDER OBJETO (Janela 11). Feito isto, aperte a tecla ESC. [br][/*][*]Ative a ferramenta CÍRCULO DEFINIDO PELO CENTRO E UM DE SEUS PONTOS (Janela 6), clique no ponto Incentro e posteriormente no ponto D. Uma circunferência h será criada.[/*][/list]
Reflexão 1
Altere as posições dos pontos A, B ou C. Por que os lados do triângulo tangenciam o círculo?
Reflexão 2
Meça as distâncias de um dos vértices do triângulo a dois pontos de tangência e observe que são iguais. Por que?[br][br] [img 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[/img]
Propriedade
As três bissetrizes internas de um triângulo interceptam-se num mesmo ponto que equidista dos 3 lados do triângulo.
Demonstração
Circuncentro
Circuncentro
O circuncentro de um triângulo é o ponto de encontro das mediatrizes dos lados do triângulo.
Construção do Circuncentro do triângulo
[list][*]Ative a ferramenta POLÍGONO (Janela 5) e clique em três lugares distintos para formar um triângulo. Para fechar o triângulo clique novamente no primeiro ponto. Naturalmente que os pontos não podem estar alinhados. Um triângulo com vértices nos pontos A, B e C será criado.[/*][*]Ative a ferramenta MEDIATRIZ (Janela 4) e clique sobre o lado c (que une o ponto A ao B), uma reta d aparecerá; clique sobre o lado b, uma reta e aparecerá. [br][/*][*]Selecione a ferramenta INTERSEÇÃO DE DOIS OBJETOS (Janela 2) e clique sobre as retas d e e. O ponto D será criado. [br][/*][*]Do mesmo modo que na atividade anterior, perguntamos: a outra mediatriz também passará pelo ponto D ? Para checar isto, novamente com a ferramenta MEDIATRIZ (Janela 4) ativada, clique sobre o lado a do triângulo. Uma reta f será criada. Verifique se esta última reta também passou pelo ponto D. [br][/*][/list]
Construção do círculo circunscrito
[list][*]Ative a ferramenta EXIBIR/ESCONDER OBJETO (Janela 12), clique sobre as retas d, e, f e aperte ESC posteriormente. [br][/*][*]Vamos modificar o nome do ponto D para Circuncentro. Para tal, clique com o botão do lado direito do mouse sobre o ponto D e selecione a opção RENOMEAR. Na nova janela que aparecerá escreva Circuncentro e clique em OK. [br][/*][*]Ative a ferramenta CÍRCULO DEFINIDO PELO CENTRO E UM DE SEUS PONTOS (Janela 6), clique no ponto Circuncentro e, posteriormente, em um dos vértices do triângulo. Uma circunferência g será criada. O que você observa?[/*][/list]
Reflexão 1
Por que o círculo passa pelos três vértices do triângulo?
Reflexão 2
Quando o circuncentro será interno ao triângulo? Quando será externo? Quando estará sobre um dos lados do triângulo?
Propriedade
As mediatrizes dos lados de um triângulo se interceptam num mesmo ponto que equidista dos três vértices.
Demonstração
Reflexão
Entendeu a demonstração?
Ortocentro
Ortocentro
O ortocentro de um triângulo [color=#000000]é o ponto de encontro das três alturas do triângulo[/color]. Também pode ser considerado como ponto de encontro das retas suportes das alturas do triângulo.
Construção do Ortocentro
Ortocentro
Reflexão 1
Quando o ortocentro estará interno ao triângulo? Quando estará externo? Quando coincidirá com um dos vértices?
Propriedade
As três retas suportes das alturas de um triângulo se interceptam num mesmo ponto.
Reflexão
Se o ortocentro estivesse externo ao triângulo, a demonstração seria a mesma?
Ângulo formado pelas alturas
Reflexão 3
Altere os vértices do triângulo anterior e observe o angulo formado pelas alturas. Quando esse ângulo será agudo? Quando ele será obtuso?
Baricentro
Baricentro
O baricentro de um triângulo [color=#000000]é o ponto de encontro das medianas do triângulo[/color]. A mediana é [color=#000000]o segmento de reta que une um vértice ao ponto médio do lado oposto[/color].
Baricentro do triângulo
[list][*]Selecione a ferramenta POLÍGONO (Janela 5) e clique em três lugares distintos para formar um triângulo. Para fechar o triângulo clique novamente no primeiro ponto. Naturalmente que os pontos não podem estar alinhados. Um triângulo com vértices nos pontos A, B e C será criado como o mostrado na figura seguinte. [/*][*]Ative a ferramenta PONTO MÉDIO OU CENTRO (Janela 2) e clique sobre o lado c. Um ponto D será criado. [/*][*]Ative a ferramenta SEGMENTO DEFINIDO POR DOIS PONTOS (Janela 3), clique no ponto C e posteriormente no ponto D. Um segmento d será criado. Esse segmento criado é chamado de mediana. [/*][*]Ative a ferramenta PONTO MÉDIO OU CENTRO (Janela 2), novamente e clique sobre o lado a. Um ponto E será criado. [/*][*]Ative a ferramenta SEGMENTO DEFINIDO POR DOIS PONTOS (Janela 3), clique no ponto A e posteriormente no ponto E. [/*][*]Ative a ferramenta INTERSEÇÃO DE DOIS OBJETOS (Janela 2), clique sobre o segmento d e, posteriormente, sobre o segmento e. Um ponto F será criado. A próxima mediana também passará por F? Para observar, ative a ferramenta PONTO MÉDIO OU CENTRO (Janela 2) novamente e clique sobre o lado b. Um ponto G será criado. [/*][*]Ative a ferramenta SEGMENTO DEFINIDO POR DOIS PONTOS (Janela 3), clique no ponto B e, posteriormente, no ponto G. Esse segmento criado é uma outra mediana. Se tudo correu bem, estará com um desenho semelhante ao que está a seguir. O ponto F é o baricentro.[br]Vamos modificar o nome do ponto F para baricentro. Para tal, clique com o botão do lado direito do mouse sobre o ponto F e selecione a opção RENOMEAR. Na nova janela que aparecerá, escreva baricentro e clique em OK. [br][/*][/list]
Reflexão 1
Use a ferramenta DISTÂNCIA, COMPRIMENTO OU PERÍMETRO (Janela 8). Meça as distâncias de A até E e de A até o baricentro. O que você observa? Meça também a distância do baricentro ao ponto E. O que você observa?[br][br]
Propriedade
As três medianas de um triângulo interceptam-se num mesmo ponto que divide cada mediana em duas partes tais que a parte que contém o vértice é o dobro da outra.
Demonstração
A reta de Euler e o centro de massa do triângulo
Reta de Euler
Construa os pontos notáveis baricentro, circuncentro e ortocentro de um mesmo triângulo ABC. Construa uma reta que passa pelo ortocentro e o circuncentro.
Reflexão sobre a reta de Euler
O baricentro pertence a reta que passa pelo o ortocentro e o circuncentro?
Propriedade do baricentro
Construa o baricentro G de um triângulo ABC. Após isso, construa os triângulos ABG, BCG e ACG. Meça a área de cada um deles. O que você observa?
Justificativa das propriedades
Pesquise e apresente fontes que mostrem as justificativas das propriedades vistas nos exercícios anteriores. (cole os links aqui)