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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.19 no.2 Temuco jun. 2017 


A Trigonometrical Approach to Morley's Observation

Ioannis Gasteratos1 

Spiridon Kuruklis2 

Thedore Kuruklis3 

1 Boston University, Department of Mathematics and Statistics, Boston, MA 02215, USA. e-mail:

2 Eurobank, Group Information and IT Security, 14234 Athens, Greece. e-mail:

3 Theoklitos, 17672 Kallithea, Greece. e-mail:


Simple trigonometrical arguments verify that in a triangle the trisectors, proximal to sides respectively, meet at the vertices of an equilateral triangle by showing that the length of each side is 8R times the sines of the angles between the sides of the triangle and the trisectors that determine it, where R is the radius of the circumcircle of the triangle. The 27 meeting points of the trisectors, proximal to a side, determine 18 such equilaterals, which in pairs share a vertex having two collinear sides and the third parallel. Hence these points are located 6 by 6 on three triples of parallel lines.

Keywords and Phrases: Angle trisection;, proximal trisector; triangle trisectors; Morley's theorem; Morley triangle; Morley's magic; Morley's miracle; Morley's mystery


Argumentos trigonométricos simples verifican que en un triángulo los trisectores, próximos a los lados respectivamente, se encuentran en los vértices de un triángulo equilátero mostrando que la longitud de cada lado es 8R veces los senos de los ángulos entre los lados del triángulo y los trisectores que lo determinan, donde R es el radio del circuncírculo del triángulo. Los 27 puntos de encuentro de los trisectores, próximos a un lado, determinan 18 tales equiláteros, que a pares comparten un vértice teniendo dos lados colineales y el tercero paralelo. Luego estos puntos están ubicados 6 por 6 en tres triples de líneas paralelas.

2010 AMS Mathematics Subject Classification: 00A05, 00A08


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