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A synthetic proof of de Longchamps' chain

Published online by Cambridge University Press:  24 October 2008

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Trinity College, Cambridge
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Extract

A famous chain of theorems, due originally to de Longchamps (l) and afterwards rediscovered by Pesci(2), Morley(3) and Grace(4), goes as follows:

(1) Given four lines in a plane, the four circumcentres O3 of the triangles formed by omitting each one of the lines in turn lie all on the same circle C4 with centre O4 say.

(2)Given five lines in a plane, the centres O4 of the five circles C4 obtained by omitting each of the five lines in turn lie all on the same circle C6 with centre O5 say.

And in general

(3) Given (n+1) lines in a plane, the (n+1) centres On of the circles Cn+1 formed by omitting each of the lines in turn lie all on the same circle Cn+1 with centre On+1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 3 , May 1971 , pp. 393 - 400
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

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