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The seven circles theorem revisited

Published online by Cambridge University Press:  18 June 2018

John R. Silvester
John R. Silvester*
Affiliation:
Department of Mathematics, King's College, Strand, London WC2R 2LS e-mail: jrs@kcl.ac.uk
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Extract

The circles C1, & , Cn form a chain of length n if Ci touches Ci + 1, for i = 1, & , n − 1, and the chain is closed if also Cn touches C1. A cyclic chain is a chain for which all the circles touch another circle S, the base circle of the chain. If Ci touches S at Pi, then P1, & , Pn are the base points of the chain. Sometimes there may be coincidences among the base points; in particular, if Pi = Pj, then the line PiPj should be interpreted as the tangent S to at Pi.

The seven circles theorem first appeared in [1, §3.1], and some historical details of its genesis can be found in John Tyrrell's obituary [2]. The theorem concerns a closed cyclic chain of length 6, and says that, if a certain extra condition is satisfied, then the lines P1P4, P2P5, P3P6 joining opposite base points are concurrent. Here and throughout, ‘concurrent’ should be read as ‘concurrent or all parallel’, that is, the point of concurrency might be at infinity.

Type
Articles
Information
The Mathematical Gazette , Volume 102 , Issue 554 , July 2018 , pp. 280 - 301
Copyright
Copyright © Mathematical Association 2018 

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References

1. Evelyn, C. J. A., Money-Coutts, G. B. and Tyrrell, J. A., The seven circles theorem and other new theorems, Stacey International (1974).Google Scholar
2. Laird, M. J. and Silvester, J. R., Obituary: John Alfred Tyrrell, 1932-1992, Bull. London Math. Soc. 43, pp. 401–405 (2011).Google Scholar
3. Martyn Cundy, H., The seven-circles theorem, Math. Gaz. 62, pp. 200–203 (1978).Google Scholar
4. Semple, J. G. and Kneebone, G. T., Algebraic projective geometry, Oxford (1952).Google Scholar
5. Wikipedia, Problem of Apollonius, accessed April 2018 at https://en.wikipedia.org/wiki/Problem_of_ApolloniusGoogle Scholar