Teorema de Steiner-Lehmus

é isósceles

O teorema de Steiner-Lehmus é um teorema da geometria elementar, primeiramente formulado por C. L. Lehmus e posteriormente provado por Jakob Steiner. Diz o teorema:

Qualquer triângulo com dois ângulos bissetores de comprimentos iguais é isósceles.[1]

O teorema foi mencionado pela primeira vez em 1840, em uma carta de Lehums para J. C. Sturm, na qual ele pediu por uma prova puramente geométrica. Sturm repassou a solicitação para outros matemáticos e Jakob Steiner foi um dos primeiros a providenciar uma solução. Desde então o teorema tornou-se tema bastante popular na geometria elementar, com publicações regulares de artigos sobre o teorema.[2][3]

Impossibilidade de uma prova direta

O teorema de Steiner-Lehmus pode ser demonstrado utilizando a geometria elementar, pela prova por demonstração contrária. Existem algumas controvérsias sobre se a prova "direta" é possível; supostas provas "diretas" têm sido publicadas, mas nem todos concordam que elas sejam "diretas". John Conway argumentou que não pode haver prova pela busca de igualdade porque o teorema é falso enquanto em domínio arbitrário.[4] Entretanto, até que alguém formule uma definição precisa do que seria uma prova "direta", ainda há espaço para debates.

Referências

  1. ↑ Weisstein, Eric W. «Steiner-Lehmus Theorem». MathWorld. Consultado em 5 de agosto de 2008 
  2. ↑ Coxeter, H. S. M. and Greitzer, S. L. "The Steiner-Lehmus Theorem." §1.5 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 14-16, 1967.
  3. ↑ Diane and Roy Dowling: The Lasting Legacy of Ludolph Lehmus, Manitoba Math Links -Volume II- Issue 3, Spring 2002
  4. ↑ Alleged impossibility of "direct" proof of Steiner-Lehmus theorem

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