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Conics related by a pentagram: a problem of J. E. Reeve

Published online by Cambridge University Press:  14 November 2011

A. J. Knight
A. J. Knight
Affiliation:
Mathematics Division, University of Sussex
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Synopsis

Two non-singular conies ω and α are said to be related by a pentagram if there exist pentads ofdistinct points {Oi} on ω and {Ai} on α (1 ≦ i ≦ 5) such that A1 ≡ O2O4. O3O5, A2 ≡ O3O5. O1O4, A3 ≡ O1O4. O2O5, A4 ≡ O2O5. O1O3 and A5 ≡ O1O3. O2O4. It is shown that relation by a pentagram is a poristic property; and a necessary condition on their mutual projective invariants that two nonsingular conies be so related is derived. Some ramifications are discussed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 84 , Issue 1-2 , 1979 , pp. 109 - 115
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Semple, J. G. and Roth, L.. Introduction to Algebraic Geometry (Oxford: Clarendon, 1949).Google Scholar
2Todd, J. A.. Projective and Analytical Geometry (New York: Pitman, 1946).Google Scholar
3Knight, A. J.. Some loci in S 5 associated with systems of conies in a plane. Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 93–107.CrossRefGoogle Scholar