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A simple proof of Lester’s theorem

Published online by Cambridge University Press:  01 August 2016

John Rigby
John Rigby*
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH email: rigby@cardiff.ac.uk
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Extract

Lester’s theorem (1997) states that in any scalene triangle the two Fermat points F and F' (to be defined later), the nine-point centre N, and the circumcentre O, are concyclic, and that the pair of points O,F separates the pair N, F'. (In certain geometrical situations a line is regarded as a circle of infinite radius, so that the word ‘concyclic’ includes ‘collinear’ as a special case, but here ‘concyclic’ means ‘lying on a proper circle of finite radius’.) Previous proofs of Lester’s theorem have involved advanced techniques and/or computer algebra; to quote from Ron Shail’s recent article [1],

‘Lester’s original computer-assisted discovery and proof make use of her theory of “complex triangle coordinates” and “complex triangle functions”. ... 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.’

Type
Articles
Information
The Mathematical Gazette , Volume 87 , Issue 510 , November 2003 , pp. 444 - 452
Copyright
Copyright © The Mathematical Association 2003

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References

1. Shail, R., A proof of Lester’s Theorem, Math. Gaz. 85 (July 2001) pp. 226–232.CrossRefGoogle Scholar
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