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.

See also

• Seven circles theorem
• Pascal theorem

References