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Extension of a set of theorems in circle geometry

Published online by Cambridge University Press:  24 October 2008

J. H. Grace
J. H. Grace
Affiliation:
Peterhouse.
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Extract

The fascinating chain of theorems due to Clifford and recently extended by Mr F. P. White originates in the elementary fact that the circumscribing circles of the triangles formed by four lines meet in a point. From a like simple germ, namely the fact that the centres of the above-mentioned circles lie on a circle, an infinite chain of theorems was first evolved by Pesci and may be enunciated as follows:

(i) The centres of the circumcircles of the triangles formed by four lines lie on a circle; thus from four lines we derive a point, namely the centre of the latter circle.

(ii) Five lines give five sets of four and the five derived points lie on a circle; thus from five lines we derive a point, namely the centre of the latter circle.

(iii) Six lines give six sets of five and the six derived points lie on a circle.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 24 , Issue 1 , January 1928 , pp. 10 - 18
Copyright
Copyright © Cambridge Philosophical Society 1928

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References

* Collected Papers, pp. 38–54.Google Scholar

† Proceedings Camb. Phil. Soc. vol. XXII, p. 684. The extensions are of the kind in which a curve in two dimensions is replaced by a curve in higher dimensions. Other extensions are alluded to below, §§ 8–11.Google Scholar

‡ Periodico di Matematica 5 (1891).Google ScholarThe results were discovered independently by Morley, , Transactions Amer. Math. Soc. 1 (1900), 97–115.Google ScholarCf. also Grace, , Proc. Lond. Math. Soc. (1) 33 (1901), 193–7.Google Scholar

* It is proved, § 9, that these points do not give rise to other and similar chains of theorems: the point in which meet the four conies derived from four lines is of coarse the first point in the Clifford chain.Google Scholar

* For n = 3 the result has already been used in § 4.Google Scholar

* Cf. White, , loc. cit. §§ 2, 3.Google Scholar

* Crelle, , CXIX (1898), 186. Cf.Google ScholarGrace, , Trans. Camb. Phil. Soc. XVI (1898), 163.Google Scholar

† This has also been remarked by Baker, : Proc. Camb. Phil. Soc. XXII (1923). 28–33.Google Scholar

* It will suffice to take a ease in which the quadric is two planes, when the result is obvious. I owe this remark to MrFraser, P..Google Scholar

† The ten faces are such that all quadrics touching nine of them touch the other; it follows that there is a linear relation between the squares of the equations of the ten primes and thence that the two simplexes are self-conjugate for the same quadric. Reciprocation with respect to this leads to the result.Google Scholar

* Or by projecting the whole figure from K on the plane π.Google Scholar