An extension of Pascal’s theorem

Abstract

In 1825 the Academie Royale de Bruxelles proposed as a prize topic the extension of Pascal's theorem to space of three dimensions. The prize was won by Dandelin,t who showed that a skew hexagon formed of three lines from each regulus of a hyperboloid of revolution has the Pascal property that pairs of opposite planes meet on a plane, and the dual, or Brianchon property, that the lines joining pairs of opposite vertices meet in a point. Hesset wrote several papers on the IDandelin skew hexagons, emphasizing the polar properties of the Pascal plane and the Brianchon point with respect to the quadric bearing the two reguli. Several of the older analytic geometries of three dimensions devote some space to the problem of the skew hexagon, as Salmon, and more notably Plucker?, who offered much original material. In a recent article, the present writerll has discussed the skew hexagon from an elementary analytic approach. The foregoing citations exhibit the idea that Pascal's theorem deals with six elements of a quadratic curve, and that the space extension offered will deal with six elements of a quadric surface. It happens that an extension of this sort is valid in space of three dimensions; but in S., where n >3, it is clear that a skew hexagon of six rulings of a hyperquadric must lie in an S3 if it is to possess the Brianchon property. It seems, therefore, that the hexagon idea must be abandoned in the search for a valid extension of Pascal's theorem in S, In the plane, a variant of Pascal's theorem affirms that if two triangles are in homology, then the six points of intersection of the sides of the one with the non-corresponding sides of the other lie upon a conic. From a consideration of the converses of this theorem and its dual it occurred in-