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A note on Morley's trisector theorem

Published online by Cambridge University Press:  24 October 2008

H. Lob
H. Lob
Affiliation:
The Grammar SchoolManchester
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Extract

Two theorems by the late Prof. F. Morley have aroused special interest, one being the chain of circle properties discovered by De Longchamps and independently by Pesci, Grace and by Morley himself, and the other the remarkable property that if the angles of any triangle are trisected, then the trisecting lines meet in pairs to form an equilateral triangle. In a previous paper‡ the present writer has shown that the circles making up the chain may be derived from a rational normal Cn in [n] by the following process.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 36 , Issue 4 , October 1940 , pp. 401 - 413
Copyright
Copyright © Cambridge Philosophical Society 1940

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References

* See papers by Richmond, H. W. in J. London Math. Soc. 14 (1939), 73, 78CrossRefGoogle Scholar, where the history of the chain is given. Richmond shows that the chain can be derived as a special case of a more general theorem. See also Richmond, , “An extension of Morley's chain of theorems on circles”, Proc. Cambridge Phil. Soc. 29 (1933), 165CrossRefGoogle Scholar, and Bath, F., “On circles determined by five lines in a plane”, Proc. Cambridge Phil. Soc. 35 (1939), 518.CrossRefGoogle Scholar

† Several proofs of the basic theorem are given in the Math. Gazette for 1922, including an elegant geometrical one by J. M. Child. Lob and Richmond deal with the question of the 27 triangles: “A neglected principle of elementary trigonometry”, Proc. London Math. Soc. 31 (1929), 355.Google Scholar

‡ “Some chains of theorems derived by successive projection”, Proc. Cambridge Phil. Soc. 29 (1933), 45.Google Scholar

* Lob and Richmond, loc. cit.