Conics and Quadrics - Main Theorem

TheoremPascal's Theorem

If six points $A, B, C, D, E, F$ lie on a conic (including degenerate conics), then the three points:

$P = AB \cap DE$
$Q = BC \cap EF$
$R = CD \cap FA$

are collinear. The line containing these points is called the Pascal line of the hexagon.

This remarkable theorem, discovered by Blaise Pascal at age 16 (1639), unifies numerous special cases. For different orderings of the six points, we obtain different hexagons and different Pascal lines—a single set of six points on a conic determines 60 distinct Pascal lines.

The theorem works for any conic, including degenerate cases (pairs of lines). When the conic degenerates to two lines, Pascal's theorem reduces to Pappus' theorem, showing that Pappus is a special case of this more general result.

DefinitionMystic Hexagram

Pascal called the configuration formed by six points on a conic and their Pascal line the mystic hexagram (hexagrammum mysticum). The rich structure includes:

6 vertices (points on the conic)
6 sides (connecting consecutive points)
3 diagonal points (Pascal line points)

This configuration has 60 different realizations for each set of six points.

ExampleSpecial Cases

When points on the conic coalesce, Pascal's theorem yields results about tangents:

Five distinct points, one double: If $B = B'$ (two points coincide), the line $BB'$ becomes the tangent at $B$ . The theorem still holds with this interpretation.

Four distinct points, two double: Yields the theorem about tangents from two points.

Circle case: For six points on a circle, Pascal's line has special angle properties related to the inscribed hexagon.

The dual of Pascal's theorem is Brianchon's theorem: if a hexagon is circumscribed around a conic, its three main diagonals (connecting opposite vertices) are concurrent. By projective duality, any statement about points on a conic has a dual statement about tangent lines to the conic.

Remark

Pascal's theorem is fundamental in algebraic geometry. It provides a purely geometric criterion for six points to lie on a conic (without solving equations). This geometric approach complements algebraic methods and reveals deeper structure.

The theorem also has connections to:

Elliptic curves: Six points on a cubic satisfy similar incidence relations (Cayley-Bacharach theorem)
Desargues' theorem: Can be derived as a limiting case of Pascal
Combinatorics: The 60 Pascal lines form interesting geometric configurations

In projective space over finite fields, Pascal's theorem provides combinatorial information about Galois geometry. The incidence structure of points and lines on conics over $\mathbb{F}_q$ has applications in coding theory and finite geometry.