Brianchon's theorem: What is it?
A Mathematical Droodle
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Copyright © 19962018 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

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Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny