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**Up:** *Projective Conics*

# Brianchon's Theorem

Another way of stating Pascal's theorem is: If points *ABCDEF*
lie on a conic, then points *AB.DE*, *BC.EF*, and
*CD.FA* lie on one line. This shows the way to a dual theorem,
known as *Brianchon's theorem:* if lines abcdef lie on a conic,
then lines *(a.b)(d.e)*, *(b.c)(e.f)*, *(c.d)(f.a)* lie
on one point. If six lines lie on a conic, then the hexagon which
they form circumscribes the conic. So the more traditional form of
Brianchon's theorem is:
If a hexagon is circumscribed about a conic, its diagonals
are concurrent.

Brianchon's theorem has a converse, too:
If the diagonals of a hexagon are concurrent, then the
hexagon may be circumscribed about a conic.

The proofs for these are exactly the duals of the proofs
for Pascal's theorem and its converse.

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**Up:** *Projective Conics*

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Created: Nov 30 1995 ---
Last modified: Thu Nov 30 15:30:47 1995