Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T18:25:25.524Z Has data issue: false hasContentIssue false

A proof of Lester’s Theorem

Published online by Cambridge University Press:  01 August 2016

Ron Shail*
Affiliation:
Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH

Extract

The object of this note is to draw readers’ attention to a very new theorem in the Euclidean geometry of the triangle and to provide a straightforward Cartesian proof. This remarkable theorem, due to Lester, asserts that in any scalene triangle the two Fermat points, the nine-point centre and the circumcentre are concyclic. Lester’s original computer-assisted discovery and proof make use of her theory of ‘complex triangle coordinates’ and ‘complex triangle functions’ as expounded in, and . A proof has also been given by Trott using the advanced concept of Grobner bases in the reduction of systems of polynomial equations to ‘diagonal’ form. Trott’s work uses the computer algebra system Mathematica as an essential tool and he also provides an animation of the Lester circle as one vertex of the triangle is varied. Further information on the configuration is given in.

Type
Articles
Copyright
Copyright © The Mathematical Association 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Lester, J. A. Triangles III: Complex triangle functions, Aequationes Mathematicae 53 (1997) pp. 435.Google Scholar
2. Lester, J. A. Triangles I: Shapes, Aequationes Mathematicae 52 (1996) pp. 3054.Google Scholar
3. Lester, J. A. Triangles II: Complex triangle coordinates, Aequationes Mathematicae 52 (1996) pp. 215245.Google Scholar
4. Trott, M. Applying GroebnerBasis to three problems in geometry, Mathematica in Education and Research 6 (1997) pp. 1528.Google Scholar
5. Fowler, D. Farewell to Greek-bearing GIF’s Part II: Java geometry, Maths and Stats 10 (1999) pp. 911.Google Scholar
6. Wells, D. The Penguin book of curious and interesting geometry, Penguin Books (1991).Google Scholar
7. Durell, C. V. Modern geometry, Macmillan (1938).Google Scholar
8. Maxwell, E. A. Geometry for advanced pupils, Oxford University Press (1953).Google Scholar