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Sums of squares of edge-lengths of cyclic polygons

Published online by Cambridge University Press:  01 August 2016

John Rigby
John Rigby*
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, emails: rigby@cardiff.ac.uk; rigbycathedralcourt@tiscali.co.uk
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Extract

Problem 87.I in a recent Gazette concerns cyclic quadrilaterals.

Let A, B, C and D be distinct points on a circle with radius r. Show that

When does equality occur?

We shall investigate generalisations of the problem by considering cyclic polygons and also points on a sphere, by distinguishing between sides and diagonals, and by looking at minima as well as maxima. The term ‘edge’ in the title is intended to encompass both sides and diagonals. It is interesting to see the wide variety of methods needed to solve the generalised problems.

Type
Articles
Information
The Mathematical Gazette , Volume 90 , Issue 517 , March 2006 , pp. 25 - 39
Copyright
Copyright © The Mathematical Association 2006

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References

1. Problem 87.1, Math. Gaz. 87 (November 2003) p. 587.Google Scholar
2. Solution to Problem 87.1, Math. Gaz. 88 (July 2004) pp. 324–325.Google Scholar
3. Bornsztein, Pierre, Crux Mathematicorum (2000) p. 276.Google Scholar
4. Lord, Nick, An unusual minimax problem, Octagon (Romania), April 1995, pp. 39–41.Google Scholar