Pascal's Theorem Research Articles

Schliessungss�tze in Laguerre-Ebenen

For Laguerre planes Artzy [1] showed that a 4-point Pascal theorem on an oval leads to a configuration π, consisting of six points and five regular circles, which is equivalent Miquel's theorem. We represent some similar incidence assumptions which are again equivalent to π in each Laguerre plane. Besides, a uniform denotation to characterize different kinds of miquelian theorems in Benz planes is suggested.

A generalisation of Pascal's theorem to commutative rings

A generalisation of Pascal's theorem to commutative rings

A symmetry theorem for Laguerre planes

A 4-point Pascal theorem on an oval in a protective plane led to the symmetry theorem in the Minkowski plane, and this relation was used by the author in [2,3] to prove that the symmetry theorem is equivalent to Miquel's theorem and that it implies the tangency theorem (Beruhrsatz). The same special Pascal theorem now leads to an incidence theorem, π, for the Laguerre plane, which is again equivalent to Miquel's theorem. The configuration of π contains 6 points and 5 circles and is thus simpler than that of Miquel, although it does not have the intuitive symmetry properties of the corresponding configuration in the Minkowski plane.

Hypersurfaces Fixed by Equiaffinities

In this note we state and prove the followingAny equiaffinity acting on the points of an n-dimensional vector space (n ≥2) leaves invariant the members of a one parameter family of hypersurfaces defined by polynomials p(xl…,xn)=c of degree m ≤n.The theorem, restricted to the real plane, appears to have been discovered almost simultaneously by Coxeter [4] and Komissaruk [5]. The former paper presents an elegant geometric argument, showing that the result follows from the converse of Pascal's theorem. The present approach is more closely related to that of [5], in which the transformations are reduced to a canonical form.

A pascal theorem applied to Minkowski geometry

If on an oval in a projective plane a 4-point Pascal theorem, π, with fixed points U and V holds, then the oval is {(x,y) ¦xy=c} ∪ (O) ∪ (∞), with c ≠ O, in some Hall coordinatization. If for every 3 distinct points P, Q, R (not on UV; neither U nor V collinear with two of P, Q, R) there is through them a certain point set satisfying an extended version of π, then all these sets together with all lines not through U or V form the circles of a plane Minkowski (= pseudoeuclidean) geometry over a commutative field. π may be expressed in terms of Minkowski geometry. Together with incidence axioms derived from the protective incidence axioms, the Minkowski version of π characterizes the plane Minkowski geometry over a commutative field and is thus equivalent to Miquel's theorem.

Addendum to “A pascal theorem applied to minkowski geometry”

The results of the author's previous paper imply that the class of all (B*S)-geometries of Benz (see Benz, Leissner, and Schaeffer [2])coincides, up to isomorphisms, with the class of all plane Minkowski geometries over commutative fields.

Perspective simplexes and tucker quadrics

The sides of a triangle are met by the sides of a triangle in perspective with it in nine points of which three lie on the axis of perspectivity of the triangles and the other six, as we have it from the converse of Pascal's theorem, lie on a conic. When the centre of perspectivity is the Lemoine point of the triangle and the axis of perspectivity is the line at infinity this conic is a circle known as a Tucker circle [1]. In this paper, analogously we prove that given a simplex inn-space, the traces on its edges of the prime faces of any simplex in perspective with it, all except those falling on the prime of perspectivity, lie on a quadric which we call a Tucker quadric. And further we consider the special case when this quadric becomes a hypersphere.

Specializations of Pascal's theorem on an oval

An oval C In a projective plane is a set of triply noncollinear points such that each point of C is on exactly one line which contains no other point of C. An oval has the Pascalian property if each of its hexagons has collinear diagonal points. The author considers those hexagons in which at least two vertices coincide and shows that if every hexagon in this class of 5-point hexagons is Pascalian then the oval has the Pascalian property. Therefore, the 5-point Pascalian property is equivalent to the full Pascalian property. The proof makes use of the coordinatization found in Artzy [1, 2].

Pascal's Theorem on an Oval

Pascal's Theorem on an Oval

Pascal's Theorem on an Oval

Pascal's Theorem on an Oval

Pascal's theorem in n–space

An analogue in a solid of the well known Pascal's theorem (Baker, [1], p. 219) for a conic is established by Baker ([2], pp. 53–54, Ex. 15) after Chasles [6] and by Salmon ([2], p. 142). The same is discussed in detail by Court [8]. The purpose of this paper is to extend it to a projective space of n dimensions or briefly to an n-space Sn. To prove it, we introduce here once again the idea of a set of n+1 associated lines in Sn as indicated in an earlier work (Mandan, [12]) in analogy with a set of 5 associated lines in S4 (Baker, [4], p. 122), and make use of the method of induction.

Various Proofs of Pascal's Theorem

(1965). Various Proofs of Pascal's Theorem. Mathematics Magazine: Vol. 38, No. 1, pp. 22-30.

Various Proofs of Pascal's Theorem

Various Proofs of Pascal's Theorem

Another Generalization of Pascal's Theorem

Another Generalization of Pascal's Theorem

Note on “The Extension of Pascal's Theorem”

Note on “The Extension of Pascal's Theorem”

A Generalization of Pascal's Theorem

A Generalization of Pascal's Theorem

The Extension of Pascal's Theorem

The Extension of Pascal's Theorem

The Extension of Pascal's Theorem

The Extension of Pascal's Theorem

An Extension of Pascal's Theorem

An Extension of Pascal's Theorem

An Extension of Pascal's Theorem

Journal of the London Mathematical SocietyVolume s1-35, Issue 3 p. 337-343 Notes and papers An Extension of Pascal's Theorem D. W. Babbage, D. W. Babbage Magdalene College CambridgeSearch for more papers by this author D. W. Babbage, D. W. Babbage Magdalene College CambridgeSearch for more papers by this author First published: July 1960 https://doi.org/10.1112/jlms/s1-35.3.337AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volumes1-35, Issue3July 1960Pages 337-343 RelatedInformation