brianchon's theorem is true in both the 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.