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On cardioids and Morley's theorem

Published online by Cambridge University Press:  17 February 2021

John R. Silvester*
Affiliation:
Department of Mathematics, King's College, Strand, London WC2R 2LS e-mail: [email protected]

Extract

Morley’s trisector theorem says that the three intersections of the trisectors of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle. See Figure 1. Morley discovered his theorem in 1899, and news of it quickly spread. Over the years many proofs have been published, by trigonometry or by geometry, but a simple angle-chasing argument is elusive. See [1] for a list up to 1978. Perhaps the easiest proof is that of John Conway [2], who assembles a triangle similar to the given triangle by starting with an equilateral triangle and surrounding it by triangles with very carefully chosen angles.

Type
Articles
Copyright
© The Mathematical Association 2021

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References

Oakley, Cletus O. and Baker, Justine C., The Morley trisector theorem, Amer. Math. Monthly 85(9) (November 1978) pp. 737745.10.1080/00029890.1978.11994688CrossRefGoogle Scholar
Conway, John, On Morley's trisector theorem, Math. Intelligencer 36(3) (Septenber 2014) p. 3, available online at https://link.springer.com/content/pdf/10.1007/s00283-014-9463-3.pdfCrossRefGoogle Scholar
Morley, Frank and Morley, F. V., Inversive geometry, Bell, London (1933) pp. 239244.Google Scholar
Craats, Jan van de and Brinkhuis, Jan, Cardioids and Morley's trisector theorem, Nieuw Arch. Wiskd. (5) 18, No. 1 (March 2017) pp. 25-33, available online at http://www.nieuwarchief.nl/serie5/pdf/naw5-2017-18-1-025.pdfGoogle Scholar