# Brianchon's theorem

In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1783–1864).

## Formal statement

Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then lines AD, BE, CF (extended diagonals each connecting opposite vertices) intersect at a single point.[1]:p. 218[2]

## Connection to Pascal's theorem

The polar reciprocal and projective dual of this theorem give Pascal's theorem.

## In the affine plane

Brianchon's theorem is true in both the affine plane and the real projective plane. However, its statement in the affine plane is in a sense less informative and more complicated than that in the projective plane. Consider, for example, five tangent lines to a parabola. These may be considered sides of a hexagon whose sixth side is the line at infinity, but there is no line at infinity in the affine plane. In two instances, a line from a (non-existent) vertex to the opposite vertex would be a line parallel to one of the five tangent lines. Brianchon's theorem stated only for the affine plane would therefore have to be stated differently in such a situation.

The projective dual of Brianchon's theorem has exceptions in the affine plane but not in the projective plane.

## Proof

Brianchon's theorem can be proved by the idea of radical axis or reciprocation.