Brianchon's theorem: What is it?
A Mathematical Droodle

Brianchon theorem is the dual of Pascal's theorem. It asserts that in a hexagon circumscribed about a conic the major diagonals, i. e. the diagonals joining vertices with the opposite ones, are concurrent.

The above applet demonstrates the theorem only for the case of the hexagon circumscribed about a circle. Any other conic section can be obtained from a circle by a projective mapping which preserves line concurrency. (However, there is also an illustration of the validity of the theorem in an arbitrary ellipse.)

It's interesting to observe how Brianchon's theorem implies theorems about pentagons and quadrilaterals. For example, the theorem of a circumscribed quadrilateral is just a particular case of Brianchon's in which two pairs of points coalesce.

The easiest way to prove Brianchon's theorem is by way of duality implied by the properties of poles and polars. Connect the successive points where the sides of the given hexagon touch the conic. The result is a second hexagon, now inscribed into the conic. Its sides lie on the polars of the vertices of the given hexagon. On the other hand, the vertices of the inscribed hexagon serve as poles of the sides of the given one. The diagonals of the circumscribed hexagon are the polars of the points of intersection of the extended opposite sides of the inscribed hexagon. The latter are collinear by Pascal's theorem. Therefore, the diagonals at hand are concurrent. This argument is obviously reversible. Therefore, Brianchon's theorem implies that of Pascal.

Poles and Polars

Pascal and Brianchon Theorems

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Brianchon's theorem: What is it? A Mathematical Droodle

Poles and Polars

Pascal and Brianchon Theorems

Brianchon's theorem: What is it?
A Mathematical Droodle