##  [Pascal's Theorem](/pascals-theorem-0) 

 Definition

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

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

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

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

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

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

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

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

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.