Editing Brianchon's theorem (section)

==In the affine plane==
Brianchon's theorem is true in both the [[Euclidean plane|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.