Editing Pascal's theorem (section)

== Euclidean variants ==

The most natural setting for Pascal's theorem is in a [[projective plane]] since any two lines meet and no exceptions need to be made for parallel lines. However, the theorem remains valid in the Euclidean plane, with the correct interpretation of what happens when some opposite sides of the hexagon are parallel.

If exactly one pair of opposite sides of the hexagon are parallel, then the conclusion of the theorem is that the "Pascal line" determined by the two points of intersection is parallel to the parallel sides of the hexagon. If two pairs of opposite sides are parallel, then all three pairs of opposite sides form pairs of parallel lines and there is no Pascal line in the Euclidean plane (in this case, the [[line at infinity]] of the extended Euclidean plane is the Pascal line of the hexagon).