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On Pascal ovals

Part of: Geometry Linear incidence geometry Finite geometry and special incidence structures

Published online by Cambridge University Press:  09 April 2009

Olga Fernandes
Olga Fernandes
Affiliation:
110 St. Patrick's Town Sholapur Road Poona 411013, India
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Abstract

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In this paper we prove that if an oval in a finite projective plane of order n ≡ 3 (mod 4) has the four point Pascal property and if each of its tangents and secants has the five point Pascal property, then the plane is Pappian and the oval is a conic.

We also establish results concerning Ostrom conics and ovals with the three point and four point Pascal properties.

MSC classification

Secondary: 51E15: Affine and projective planes 51A25: Algebraization
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 1 , February 1993 , pp. 61 - 69
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Artzy, R., ‘Pascal's theorem on an oval’, Amer. Math. Monthly 75 (1968), 143–146.Google Scholar
[2]Bruck, R. H., A survey of binary systems (Springer, Berlin, 1958).CrossRefGoogle Scholar
[3]Buekenhout, F., ‘Plans projectifs à ovoïdes Pascaliens’, Arch. Math. 17 (1966), 89–93.CrossRefGoogle Scholar
[4]Fernandes, O., ‘Harmonic points and the intersections of ovals’, Geom. Dedicata 19 (1985), 271–276.CrossRefGoogle Scholar
[5]Fernandes, O., ‘On an oval with the four point Pascalian property’, Canad. Math. Bull. 27 (1984), 295–300.Google Scholar
[6]Hofmann, C. E., ‘Specializations of Pascal's theorem on an oval’, J. Geom. 1 (1971), 143–153.CrossRefGoogle Scholar
[7]Korchmáros, G., ‘Una generalizzazione del teorema di F. Buekenhout sulle ovali pascaliane’, Boll. Un. Mat. Ital. 18 (1981), 673–687.Google Scholar
[8]Ostrom, T. G., ‘Conicoids: conic-like figures in non-Pappian planes’, in: Geometry-von Staudt's point of view (eds. Plaumann, P. and Strambach, K.) (D. Reidel, Dordrecht, 1981) pp. 175–196.CrossRefGoogle Scholar
[9]Ostrom, T. G., ‘Ovals, dualities and Desargue's theorem’, Canad. J. Math. 7 (1955), 417–431.Google Scholar
[10]Rigby, J. F., ‘Pascal ovals in projective planes’, Canad. J. Math. 21 (1969), 1462–1476.CrossRefGoogle Scholar
[11]Segre, B., ‘Ovals in a finite projective plane’, Canad. J. Math. 7 (1955), 414–416.CrossRefGoogle Scholar