Abstract: Pascal's Theorem states that if a hexagon is inscribed in a circle, and opposite sides are extended to meet, then the three intersection points lie on the same line. It was discovered by Blaise Pascal when he was only 16 years old. The dual fact is called Brianchon's Theorem, and states that if a circle is inscribed in a hexagon, then the main diagonals intersect at a single point.

We will see how some elementary ideas from algebraic geometry can be useful in proving these theorems, not just for a circle but for any conic section, i.e. ellipse, parabola, and hyperbola.