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On the Geometry of the Conic and Triangle
Published online by Cambridge University Press: 20 January 2009
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In the Proceedings of 1905–6 Mr Pinkerton gave an extension of the nine point circle to a nine point conic. This raises the question of the extension of the geometry of the circle and triangle to that of the conic and triangle. If a triangle with its associated system of lines and circles be orthogonally projected on a second plane we have a triangle with an associated system of lines and homothetic ellipses. Pairs of perpendicular lines are projected into lines parallel to pairs of conjugate diameters. Such lines will be called, for shortness, in the sequel, conjugate lines. In any relation between lengths of lines, these lengths will be replaced by their ratios to the lengths of the parallel radii of one of the homothetic ellipses.
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- Copyright © Edinburgh Mathematical Society 1907
References
* I am indebted to Dr Muirhead for reference to a paper by L. Ripert (La Dualité et l'Homographie dans le Triangle et le Tétrah`dre, Paris, Gauthier-Villars et File, 1898.) In this short generalising paper M. Ripert has anticipated some of my fundamental ideas. His work is, however, very different, being mainly an application of barycentric co-ordinates. I may state, that in the detailed working out of results, I often used proofs by trilinear and barycentric co-ordinates, but discarded them for proofs involving the direct application of the principles already mentioned.
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