Trisectors like Bisectors with equilaterals instead of Points

• Spiridon A Kuruklis ^[1]
  1. [1] Eurobank Risk Management Division
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 16, Nº. 2, 2014, págs. 71-110
• Idioma: inglés
• DOI: 10.4067/S0719-06462014000200005
• Enlaces
  • Texto completo
• Resumen
  • español

    Se establece que entre todos los triángulos de Morley de /\ABC, los únicos equiláteros son theones determinados por las intersecciones del proximal a cada lado de los trisectores /\ABC de ángulos interior, o exterior, o uno interior y dos exteriores. Se muestra que estos están en triángulos equiláteros de facto con demostraciones uniformes. Luego, se observa que las intersecciones de trisectores interiores con los lados de un equilátero Morley interior forman tres triángulos equiláteros. Junto con el axioma de Pasch, se utilizan para probar que el Teorema de Morley no se satisface si se usan los trisectores de un ángulo exterior y dos interiores.

  • English

    It is established that among all Morley triangles of /\ABC the only equilaterals are the ones determined by the intersections of the proximal to each side of /\ABC trisectors of either interior, or exterior, or one interior and two exterior angles. It is showed that these are in fact equilaterals, with uniform proofs. It is then observed that the intersections of the interior trisectors with the sides of the interior Morley equilateral form three equilaterals. These along with Pasch’s axiom are utilized in showing that Morley’s theorem does not hold if the trisectors of one exterior and two interior angles are used in its statement.

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