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Triangles triply in perspective

Published online by Cambridge University Press:  20 January 2009

J. A. Third
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The first part of the present paper reproduces an article contributed by me to the July issue of Mathesis, 1900, entitled “Sur les triangles trihomologiques”; the second part contains fresh developments.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , February 1900 , pp. 10 - 22
Copyright
Copyright © Edinburgh Mathematical Society 1900

References

* This case is given in Steiner's Theorie der Kegelschnitte (ed. Schröter, 1867), p. 287.

* Stated and proved by means of orthogonal projection by Prof. Neuberg in Mathenis, 2nd series, Vol. V., p. 199.

† The proposition up to this point is given in Malhesis, 1st series, Vol. II., p. 42, by Prof. Neuberg.

* and to a paper of his own which I have so far been unable to procure, read before the French Association, 1900, entitled, “Sur des groupes de triangles trihomologiques inscrits ou circonscrits a une meme conique ou a une famille de coniques.”

† This is the case in the theorem of § I., (2), (ii) if we replace Σ by Ω, and interchange A´, B´, C with S1, S2, S3; for ABC and S1S2S3 are obviously in triple perspective, A´, B´, U´ being the centres.

‡ One or two alterations in the proofs given in this part of the paper as read to the Society have been made on the suggestion of Mr R. F. Muirhead.

* See “Résumé des propriétés concernat les triangles d'aire maximum inscrits dans Pellipse,” by Barisien, M. E. N., Mathesix, 2nd series, Vol. V., p. 42.Google Scholar

† Of the many properties that may be deduced from the fact that the T' group consists of triangles of maximum area inscribed in concentric, similar and similarly situated ellipses, one of the most interesting is that all the triangles of the group have the same Brocard angle.

* This theorem is given by Rogier, M. G., in a “ Note sur les points isobariques,” Jownal de Mathématiqites Speciahs, 1887, p. 103Google Scholar

* These theorems are proved for the ease of three isobaric points and the triangle of reference by M. Rogier, loc. cit.