Brianchon's theorem

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 ${\displaystyle P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}}$ be a hexagon formed by six tangent lines of a conic section. Then lines ${\displaystyle {\overline {P_{1}P_{4}}},\;{\overline {P_{2}P_{5}}},\;{\overline {P_{3}P_{6}}}}$ (extended diagonals each connecting opposite vertices) intersect at a single point ${\displaystyle B}$, the Brianchon point.^{[1]: p. 218 [2]}

Connection to Pascal's theorem[]

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

Degenerations[]

3-tangents degeneration of Brianchon's theorem

As for Pascal's theorem there exist degenerations for Brianchon's theorem, too: Let coincide two neighbored tangents. Their point of intersection becomes a point of the conic. In the diagram three pairs of neighbored tangents coincide. This procedure results in a statement on inellipses of triangles. From a projective point of view the two triangles ${\displaystyle P_{1}P_{3}P_{5}}$ and ${\displaystyle P_{2}P_{4}P_{6}}$ lie perspectively with center ${\displaystyle B}$. That means there exists a central collineation, which maps the one onto the other triangle. But only in special cases this collineation is an affine scaling. For example for a Steiner inellipse, where the Brianchon point is the centroid.

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's theorem

References[]