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Stereometry

Published online by Cambridge University Press:  03 November 2016

Vincenzo G. Cavallaro
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Extract

In this note I give the important trigonometric relations between the Brocard and Steiner angles of the cross-section of a triangular prism and the angle which the plane of this cross-section makes with the plane cutting the prism in an equilateral triangle. To facilitate a comprehension of the whole problem, I also give a complete and brief solution of the known problem for the equilateral section of the prism, deducing the equations of the equilateral case from the equations for the general case in which the section is a triangle similar to a given triangle.

Type
Research Article
Information
The Mathematical Gazette , Volume 36 , Issue 318 , December 1952 , pp. 273 - 275
Copyright
Copyright © The Mathematical Association 1952

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References

* See Cavallaro, V.G., “On Lemoine’s Ellipse,” Math. Gazette, No. 310, p. 266 Google Scholar (December, 1950). For the Steiner angles, reference may be made to the classic Traité de Géométrie of Rouché and Comberousse, Vol. I, 8th edition, 1935, p. 477 (Gauthier-Villars, Paris), Note 111, Gdométrie du triangle, by J. Neuberg.

† See Rouché et Comberousse, loc. cit., p. 477.

‡ Cavallaro, V.G., Mathesis, 1938, No. 9–10.Google Scholar It is known (see the article by Cavallaro cited in the first footnote) that the non-focal axis 2y of Lemoine’s ellipse is given by the formula . now, by the equations (1) and (5), we have, on substitution,