Pascal's Theorem

Theorem
Let $ABCDEF$ be a hexagon whose $6$ vertices lie on a conic section and whose opposite sides are not parallel.

Then the points of intersection of the opposite sides, when produced as necessary, all lie on a straight line.

Proof

 * PascalsTheorem.png


 * PascalsTheorem2.png

Also see

 * Pappus's Hexagon Theorem
 * Desargues' Theorem