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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

REFERENCES

(1)de Longchamps, G.Note de géometrie. Nouvelle correspondance mathématique 3 (1877), 306–312 and 340–347.Google Scholar
(2)Pesci, G.Dei circoli circoscritti ai triangoli formati do n rette poste in un piano. Period. Math. 5 (1891), 120–127.Google Scholar
(3)Morley, F.On the metric geometry of the plane n−line. Trans. Amer. Math. Soc. 1 (1900), 193–197.Google Scholar
(4)Grace, J. H.On a class of plane curves. Proc. London Math. Soc. 33 (1900), 193–197.CrossRefGoogle Scholar
(5)Clifford, W. K.Synthetic proof of Miquel's theorem. Oxford, Cambridge and Dublin Messenger of Mathematics 5 (1871), 124–141;reprinted in Collected Papers, pp. 38–54 (Macmillan, London, 1882).Google Scholar
(6)Serret, P.Sur les hyperboles équilatères d'ordre quelconque. C.R. Acad. Sci. Paris 121 (1895), 340–342.Google Scholar
(7)Serret, P.Sur les faisceaux réguliers et les équilatères d'ordre n. CM. Acad. Sci. Paris, 121 (1895), 372–375.Google Scholar
(8)Grace, J. H.Extension of a set of theorems in circle geometry. Proc. Cambridge Philos. Soc. 24 (1928), 10–18.CrossRefGoogle Scholar
(9)White, F. P.An extension of Wallace's, Miquel's and Clifford's theorems on circles. Proc. Cambridge Philos. Soc. 22 (1925), 684–687.CrossRefGoogle Scholar
(10)Wallace, W. (pseudonym ‘Scoticus’). Question No. 86, Art. XXVI in The Mathematical Repository, ed. Leybourn, T., Vol. 1, New Series (1806). See Part C (1804), p. 22; also Part I (1805), pp. 169–170.Google Scholar
(11)Steiner, J.Théorème sur le quadrilatère complet. Ann. de Math. 18 (1827), 302–303.Google Scholar
(12)Miquel, A.Mérnoire de gébmetrie (2me Partie.) Angles curvilignes, intersections des cercles et des sphères. Liouville J. de Math. 10 (1845), 347–350.Google Scholar
(13)Bath, F.On circles determined by five lines in a plane. Proc. Cambridge Philos. Soc. 35 (1939), 518–519.CrossRefGoogle Scholar
(14)Baker, H. F.An introduction to plane geometry, 382 pp. (Cambridge University Press, 1943.)Google Scholar
(15)Lob, H.Some chains of theorems derived by successive projection. Proc. Cambridge Philos. Soc. 29 (1933), 45–51.CrossRefGoogle Scholar
(16)Casey, J.On bicircular quartics. Trans. Roy. Irish Acad. 24 (1871), 457–569.Google Scholar