Pascal's theorem

In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem, Latin for mystical hexagram) states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet at three points which lie on a straight line, called the Pascal line of the hexagon.

The theorem is also valid in the Euclidean plane, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel.

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.

A degenerate case of Pascal's theorem (four points) is interesting; given points ABCD on a conic Γ, the intersection of alternate sides, AB ∩ CD, BC ∩ DA, together with the intersection of tangents at opposite vertices (A, C) and (B, D) are collinear in four points; the tangents being degenerate 'sides', taken at two possible positions on the 'hexagon' and the corresponding Pascal line sharing either degenerate intersection.

If the conic is a circle, then another degenerate case says that for a triangle, the three points that appear as the intersection of a side line with the corresponding side line of the Gergonne triangle, are collinear.

The converse is the Braikenridge–Maclaurin theorem, named for 18th-century British mathematicians William Braikenridge and Colin Maclaurin (Mills 1984), which states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic; the conic may be degenerate, as in Pappus's theorem.

The theorem was generalized by August Ferdinand Möbius in 1847, as follows: suppose a polygon with 4n + 2 sides is inscribed in a conic section, and opposite pairs of sides are extended until they meet in 2n + 1 points.

A short elementary computational proof in the case of the real projective plane was found by Stefanovic (2010).

If we are to show that X = AB ∩ DE, Y = BC ∩ EF, Z = CD ∩ FA are collinear for concyclic ABCDEF, then notice that △EYB and △CYF are similar, and that X and Z will correspond to the isogonal conjugate if we overlap the similar triangles.

There also exists a simple proof for Pascal's theorem for a circle using the law of sines and similarity.

The Cayley–Bacharach theorem is also used to prove that the group operation on cubic elliptic curves is associative.

See the degenerate cases given in the added scheme and the external link on circle geometries.