**Some Definitions**

A hexagon has the Pascal Property
if the intersections of opposite sides are collinear.

A hexagon has the Brianchon Property
if the diagonals (i.e. lines joining opposite vertices) are concurrent.

Pascal's Theorem states that a hexagon
inscribed in a conic has Pascal's Property.

The Converse of Pascal's Theorem states that the vertices of
a hexagon with Pascal's Property lie on a conic.

Brianchon's Theorem states that a hexagon
circumscribed round a conic has Brianchon's Property.

**The Dual Conic Theorem**

Suppose that **C** and **D** are conics.

Then the dual of **D** with respect to **C** is a conic.

**Proof**

Throughout the proof **duality** means **duality with respect to C.**

Let A,B,C,D and E be distinct points on **D**, fixed once for all.

Then **D** is the unique conic determined by A .. E.

Let F be any other point on **D**, and

let **H** be the hexagon composed of the tangents to **D** at these six points.

By Brianchon's Theorem, the hexagon **H** has the Brianchon Property,

and so the dual hexagon **H'** has the Pascal Property.

Thus, by the Converse of Pascal's Theorem,

the vertices of **H'** lie on a conic **E**.

The vertices of **H'** correspond to A .. F (being the duals of the tangents to **D** at these points).

Since A .. E are fixed, so is the conic throught the vertices of **H'**.

As F varies on **D**, the locus of the dual of the tangent at F will be the conic **E**.