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On an extension of Wallace's pedal property of the circumcircle

Published online by Cambridge University Press:  24 October 2008

H. W. Richmond
H. W. Richmond
Affiliation:
King's College
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Extract

1. Mr J. P. Gabbatt has discussed in the most recent Part of the Proceedingsof this Society the Pedal locus of a simplex in hyperspace. It is, however, possible to regard the pedal property of the circumcircle somewhat differently and so to seek other extensions. Given a circle, any three points on it are vertices of an inscribed triangle, and the feet of the perpendiculars on the sides from any fourth point of the circle are collinear. Is there any curve in space on which an analogous property holds for any five points, viz. that the feet of the perpendiculars from any one upon the faces of the tetrahedron formed by the other four are coplanar?

It will be shown that curves of order n exist in Euclidean space of n dimensions on which any n + 2 points have such a property; but that the curves cannot be real if n is odd.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 1 , February 1924 , pp. 34 - 41
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* Proc. C.P.S. vol. XXI, part VI, p. 763.Google Scholar I am indebted to Mr Gabbatt for the two references at the end of his first paragraph.

* See Gabbatt, , l.c.. 1st paragraph for references.Google Scholar

† See Mantel, W., Wiskundige Opgaven, 1899–1902, p. 396, No. 199.Google Scholar I have to thank Mr J. H. Grace for drawing my attention to this theorem and Mantel's beautiful proof, and Mr F. P. White for the reference which I had failed to find. Theorem (i) holds for Euclidean and non-Euclidean geometry, (ii) only in Euclidean; both can be extended to n dimensions.

Another proof is due to Mr Grace. Five quadrics can be constructed, one with Aas centre self-polar to BCDE, one with B as centre self-polar to ACDE, etc.; these five are always similar and similarly situated. For if the first quadric is transformed by a homogeneous strain into a sphere, BCDEwill be strained into a tetrahedron of which A is the orthocentre. But if so B is the orthocentre of ACDE, and the second quadric must be changed into a sphere by the same homogeneous strain as the first. The condition for coplanarity of the feet of the perpendiculars is that one (and therefore each) of the quadrics should have its linear invariants equal to zero, or pass to infinity in three mutually perpendicular directions.