Brianchon's theorem

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Brianchon's Theorem.svg

In geometry, Brianchon's theorem, named after Charles Julien Brianchon (1783–1864), is as follows. Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then lines AD, BE, CF intersect at a single point.

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

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 (nor in the projective plane unless one chooses a line to play that role). A line from a vertex to the opposite vertex would then be a line parallel to one of the five tangent lines. Brianchon's theorem stated only for the affine plane would be uninformative about such a situation.

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

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