## 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.**

*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)