Geometry • Topic 34

Pascal's Theorem

Pascal's Theorem, often called the "Mystic Hexagram," describes a surprising collinearity property of points on a conic section.

Statement

Theorem (Pascal's Theorem)

Let

A BC D EF

be a hexagon inscribed in a conic section (e.g., a circle). Let the opposite pairs of sides intersect at three points:

$P = A B \cap D E$
$Q = BC \cap EF$
$R = C D \cap F A$

Then points

P, Q, R

are collinear. The line is called the Pascal Line.

Degenerate Cases

We can let vertices coincide to get theorems about tangents.

5 points: $A = B$ (Tangent at $A$ ).
4 points: Tangents at opposite vertices.

Brianchon's Theorem (Dual)

The dual of Pascal's Theorem is Brianchon's Theorem: If a hexagon is circumscribed about a conic, the three diagonals connecting opposite vertices are concurrent.

Practice Problems

Exercise (Problem 1)

Given 5 points on a circle, construct the tangent at one of the points using only a straightedge (using Pascal's theorem).

Exercise (Problem 2)

Prove that for a triangle inscribed in a circle, the tangents at the vertices intersect the opposite sides at three collinear points. (Degenerate Pascal).

Exercise (Problem 3)

Use Brianchon's theorem to prove that the diagonals of a quadrilateral with an inscribed circle intersect at a point.