Hesse's Theorem for a Quadrilateral Whose Sides Touch a Conic
Canadian mathematical bulletin, Tome 3 (1960) no. 3, pp. 221-223
Voir la notice de l'article provenant de la source Cambridge
Hesse's theorem states that “if two pairs of opposite vertices of a quadrilateral are respectively conjugate with respect to a given polarity, then the remaining pair of vertices are also conjugate ”.In the real projective plane there cannot exist such a quadrilateral, all four sides of which are self-conjugate [1, §5.54]. We shall show that such a quadrilateral exists in PG(2, 3), and that any geometry in which such a quadrilateral exists contains the configuration 134 of PG(2,3). We shall thus provide a synthetic proof of Hesse1 s theorem for a quadrilateral of this type, which, together with [1, § 5.55], constitutes a complete proof of the theorem valid in general Desarguesian projective geometry. We shall also show analytically that a finite Desarguesian geometry which admits a Hessian quadrilateral all of whose sides touch a conic must be of type PG(2, 3n).
Brown, William G. Hesse's Theorem for a Quadrilateral Whose Sides Touch a Conic. Canadian mathematical bulletin, Tome 3 (1960) no. 3, pp. 221-223. doi: 10.4153/CMB-1960-027-0
@article{10_4153_CMB_1960_027_0,
author = {Brown, William G.},
title = {Hesse's {Theorem} for a {Quadrilateral} {Whose} {Sides} {Touch} a {Conic}},
journal = {Canadian mathematical bulletin},
pages = {221--223},
year = {1960},
volume = {3},
number = {3},
doi = {10.4153/CMB-1960-027-0},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1960-027-0/}
}
TY - JOUR AU - Brown, William G. TI - Hesse's Theorem for a Quadrilateral Whose Sides Touch a Conic JO - Canadian mathematical bulletin PY - 1960 SP - 221 EP - 223 VL - 3 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1960-027-0/ DO - 10.4153/CMB-1960-027-0 ID - 10_4153_CMB_1960_027_0 ER -
Cité par Sources :