## Pascal's Theorem

**Interactive Course on Projective Geometry**

Since five points determine a conic, six points cannot be
on the same conic unless they satisfy some special condition. This
condition, discovered by the mathematician and philosopher Blaise Pascal,
is one of the earliest and prettiest results on projective geometry.

**Theorem 0.1** (Pascal's theorem)
The six vertices of a hexagon lie on a conic if and only if the three
points obtained by intersecting the three pairs of opposite sides are
collinear (Fig. 1).

Figure 1.

The following two exercises prove that if the six vertices
of a hexagon lie on a conic the three points obtained by intersecting the
three pairs of opposite sides are collinear. The proof of the converse
uses the same ideas and is left for the reader as a less structured
exercise.

Consider the transformation that takes the points of the line to
the points of the line that is defined by the following construction
(Fig.2):

- If , draw the line .
- Let be the second point of intersection of the line and
the conic.
- Draw the line .
- Let be the point
.

Figure 2.

**Exercise 0.1** (05)
Prove that the transformation

is a perspectivity.

**Exercise 0.2** (05)
Show that

and that the center of perspective is

. Verify
that this proves the first part of Pascal's theorem.

**Exercise 0.3** (10)
Show that if the three points obtained by intersecting the
three pairs of opposite sides of a hexagon are collinear, its vertices
lie on a conic.

The dual of Pascal's theorem is the following result originally dicovered
by Brianchon.

**Theorem 0.2** (Brianchon's theorem)
The six sides of a hexagon are tangent to a conic if and only if the
three lines obtained by joining the three pairs of opposite vertices are
concurrent (Fig. 3).

Figure 3.