We briefly recap Pascal's fascinating `Hexagrammum Mysticum' Theorem, and then
introduce the important dual of this result, which is Brianchon's Theorem. Essentially
Brianchon's theorem says that if one circumscribes a hexagon on any circle (or, in fact, any conic section),
and then draws lines through opposite vertices of the hexagon, then these three lines meet at a unique point.
We also discuss relationships between Pascal's line and the Brianchon point. For example, it appears as though Pascal's line is the
polar of the Brianchon point (i.e., the Appolonian dual). The depth of these results is enough by itself, but these two theorems also
reveal important truths about the nature of conics in projective geometry.
Protective geometry is deeper and more fundamental than standard euclidean geometry, and has many applications in fundamental physics, biology and perspective drawing. We shall introduce it visually, without relying upon equations. The hope is make this beautiful subject accessible to anybody, without requiring prior knowledge of mathematics. At the same time, there are some very deep, rarely discussed ideas in this subject which could also benefit experts.