Pascal's Theorem - Euclidean Variants

Euclidean Variants

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

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).