A Generalization of Cox's Chain of Theorems
Canadian mathematical bulletin, Tome 4 (1961) no. 1, pp. 1-6
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In [5, p. 105] attention has been called to a set of propositions, due to H. Cox [3, p. 67], which are related to another set, due to Clifford [2, p. 145; 4, p. 447], concerning points and circles in the plane or on the sphere. One may state Cox's chain of theorems as follows:In a projective 3-space, S3, let (1), (2), (3), (4) be four points lying in a plane α such that no three of them are collinear. Every two determine a line; let one plane such as [12], pass through each line. There are six such planes. The planes [12], [23], [13] determine a point (123); there are four such points. The first theorem of the chain states that they all lie in one plane [1234], It is not difficult to see that this is, in fact, a rewording of Möbius's theorem on mutually inscribed pairs of tetrahedra [4, p. 444].
Al-Dhahir, M. W. A Generalization of Cox's Chain of Theorems. Canadian mathematical bulletin, Tome 4 (1961) no. 1, pp. 1-6. doi: 10.4153/CMB-1961-001-9
@article{10_4153_CMB_1961_001_9,
author = {Al-Dhahir, M. W.},
title = {A {Generalization} of {Cox's} {Chain} of {Theorems}},
journal = {Canadian mathematical bulletin},
pages = {1--6},
year = {1961},
volume = {4},
number = {1},
doi = {10.4153/CMB-1961-001-9},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1961-001-9/}
}
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