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A theorem of Dr P. Zeeman

Published online by Cambridge University Press:  24 October 2008

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

The following theorem, first enunciated by Dr Zeeman in 1899, has attracted the attention of several writers:

If five points A, B, C, D, E in three dimensions are such that the feet of the perpendiculars from E on the faces of the tetrahedron ABCD are coplanar the relation between the five points is symmetrical.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 23 , Issue 8 , October 1927 , pp. 845 - 852
Copyright
Copyright © Cambridge Philosophical Society 1927

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References

* Wiskundige Opgaben, 1899–1902. Other references may be found in Mr Richmond's paper from which I take this.Google Scholar

† Proc. C.P.S., vol. XXII, 1923, p. 34. The analogue of Mantel's theorem in two dimensions was known earlier and occurs in graphical statics. Obscurities in the present exposition will be partially removed if the reader will construct the corresponding argument in two dimensions.Google Scholar

* The reader will see that the reasoning is a natural extension of what might be used to establish the corresponding property of four concyclic points.

† The argument can be alternatively expressed in quadriplanar coordinates.