A projective-geometry theorem: for any hexagon whose six vertices lie on a nondegenerate conic, the three intersection points of the pairs of opposite sides are collinear (lie on a single line called the Pascal line).

Principle

Projective invariance of incidence: incidences and collinearities determined by a conic are preserved under projective transformations; the configuration of opposite-side intersections of a hexagon inscribed in a conic yields a single line.

Demonstration

Take six distinct points A,B,C,D,E,F on a circle (a special conic). Form the hexagon A–B–C–D–E–F and compute intersections X = AB∩DE, Y = BC∩EF, Z = CD∩FA; Pascal's theorem asserts X,Y,Z are collinear. The same conclusion holds after applying any projective map sending the circle to another nondegenerate conic.

Misapplication

Asserting collinearity when the six vertices are not on a single conic (for example arbitrary hexagons) or treating degenerate placements (three consecutive vertices coincident, or a pair of opposite sides parallel in an affine picture) without handling the projective limits can produce incorrect conclusions.

Consequence

Provides a fundamental incidence relation used to generate new projective lines from conic-inscribed hexagons; it yields constructions (the Pascal line) and underlies many classical results and synthetic proofs in projective geometry.

Reversal

Dual statement (Brianchon's theorem): for a hexagon circumscribed about a conic, the three main diagonals are concurrent. Reversal highlights the projective duality between collinearity and concurrency.

Boundary

Holds in the projective plane for nondegenerate conics and properly interpreted degenerations; it does not apply if the six vertices fail to lie on a single conic or if the ambient geometry lacks the projective incidence axioms (for example arbitrary metric-only Euclidean constructions without projective closure).

Semantic Tension

Sometimes confused with purely metric hexagon properties (e.g., equal opposite sides) or with specialized circle theorems; the tension is between projective collinearity (a cross-ratio/invariance phenomenon) and Euclidean length/angle statements.

Synthesis

Pascal's theorem is a projective incidence rule: a hexagon inscribed in a conic produces a canonical line through the three intersections of opposite sides, a statement stable under projective maps and dualized by Brianchon.