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The Double Six of Lines and a Theorem in Euclidean Plane Geometry

  • John Dougall (a1)

Extract

The object of the present paper is to establish the equivalence of the well-known theorem of the double-six of lines in projective space of three dimensions and a certain theorem in Euclidean plane geometry. The latter theorem is of considerable interest in itself for two reasons. In the first place, it is a natural extension of Euler's classical theorem connecting the radii of the circumscribed and the inscribed (or the escribed) circles of a triangle with the distance between their centres. Secondly, it gives in a geometrical form the invariant relation between the circle circumscribed to a triangle and a conic inscribed in the triangle. For a statement of the theorem, see § 13 (4).

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References

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* Salmon, , Analytic Geometry of Three Dimensions, Vol. II (5th edition) §§ 534, 536a. Baker, H. F., Principles of Geometry, Vol. III, p. 159; Vol. IV, pp. 58–64.

* Salmon, Conic Sections, Chapter on Invariants and Covariants of Systems of Conies.

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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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