# Brianchon's theorem: What is it?

A Mathematical Droodle

25 March 2015, Created with GeoGebra

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Copyright © 1996-2017 Alexander Bogomolny

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

## Poles and Polars

- Poles and Polars
- Brianchon's Theorem
- Complete Quadrilateral
- Harmonic Ratio
- Harmonic Ratio in Complex Domain
- Inversion
- Joachimsthal's Notations
- La Hire's Theorem
- La Hire's Theorem, a Variant
- La Hire's Theorem in Ellipse
- Nobbs' Points, Gergonne Line
- Polar Circle
- Pole and Polar with Respect to a Triangle
- Poles, Polars and Quadrilaterals
- Straight Edge Only Construction of Polar
- Tangents and Diagonals in Cyclic Quadrilateral
- Secant, Tangents and Orthogonality
- Poles, Polars and Orthogonal Circles
- Seven Problems in Equilateral Triangle, Solution to Problem 1

### Pascal and Brianchon Theorems

- Pascal's Theorem
- Pascal in Ellipse
- Pascal's Theorem, Homogeneous Coordinates
- Projective Proof of Pascal's Theorem
- Pascal Lines: Steiner and Kirkman Theorems
- Brianchon's theorem
- Brianchon in Ellipse
- The Mirror Property of Altitudes via Pascal's Hexagram
- Pappus' Theorem
- Pencils of Cubics
- Three Tangents, Three Chords in Ellipse
- MacLaurin's Construction of Conics
- Pascal in a Cyclic Quadrilateral
- Parallel Chords
- Parallel Chords in Ellipse
- Construction of Conics from Pascal's Theorem
- Pascal: Necessary and Sufficient
- Diameters and Chords
- Chasing Angles in Pascal's Hexagon
- Two Triangles Inscribed in a Conic
- Two Triangles Inscribed in a Conic - with Solution
- Two Pascals Merge into One
- Surprise: Right Angle in Circle

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Copyright © 1996-2017 Alexander Bogomolny

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