Pascal's Theorem

Pascal's Theorem asserts: If one is given six points on a conic section and makes a hexagon out of them in an arbitrary order, then the points of intersection of opposite sides of this hexagon will all lie on a single line.

Pascal's configuration

Move the points by dragging them. Reload the page to restore the initial configuration.

There are many proofs known, but one of the most attractive is due to the nineteenth century Belgian mathematician Germinal Dandelin. It has the virtue of being perhaps the most geometric in flavour. This proof has much in common with the best known proof of Desargues' Theorem, in that it can be interpreted in terms of a three-dimensional configuration. Both Desargues' Theorem and Pascal's Theorem assert that three points defined as the intersections of certain pairs of lines in a configuration are collinear. One very natural way to see that three points in a plane are collinear is to find a plane other than the given one containing the three given points, which must all therefore lie on the intersection of two planes. For Desargues' Theorem, this plane is simple to construct, for Pascal's not quite so simple.

The three-dimensional proof of Pascal's Theorem is now well known, but seems to have originated with Dandelin.

References

F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer-Verlag, Berlin, 1973. [pp. 252-255]

H. S. M. Coxeter, The Real Projective Plane, Cambridge Press, 1955. [p. 103]

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, 1965. [pp. 102-106]

Dandelin's proof

Dandelin's three-dimensional proof of Pascal's Theorem (and its dual, Brianchon's Theorem) is contained in the paper

G. P. Dandelin, MISSING, .

We include below a translation of this paper into English, as well as a set of pictures to illustrate its argument. Although the basic argument is relatively simple and geometrical in anture, it is not so easy to illustrate it convincingly. It is interesting in this regard to examine critically the figures in Bachmann and Hilbert & Cohn-Vossen accompanying their proofs. Bachmann's figures [on p. 253 and p. 254] are schematic in nature, and while the figures in Hilbert & Cohn-Vossen [pp. 104 & 105] are somewhat more realistic, they do not make the argument as transparent as one might wish.

An English translation of Dandelin's paper

A gallery of pictures to accompany Dandelin's paper (work of Tzu-Pei Chen)