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Morley’s Triangle

Published online by Cambridge University Press:  03 November 2016

W. J. Dobbs
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Extract

I. There are three ways of trisecting an angle bounded by OA and OB ( = 3αor 3α +2π or 3α +4π), namely:

(i) OX 1 and OY 1,

(ii) OX 2 and OY 2,

(iii) OX 3 and OY 3,

as shown in Fig. 1.

It is well known that if the Euclidean angles of a triangle ABC are trisected as in Fig. 2, then the triangle xyz (commonly quoted as Morley’s Triangle) is equilateral. But angles CAB, ABC, BCA may be trisected each in three ways, so that there are really 27 “Morley” triangles to be considered. The purpose of the present paper is to show that 18 of these 27 triangles are equilateral.

Information

Type
Research Article
Information
The Mathematical Gazette , Volume 22 , Issue 248 , February 1938 , pp. 50 - 57
Copyright
Copyright © Mathematical Association 1938

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References

* I am indebted to Mr. S. W Finn for drawing my attention to these 9 equilateral triangles. They are not, of course, “Morley” triangles, but they led to the discovery of the completed Fig. 8.