Welcome #
If you’ve never encountered projective geometry before, you’re in for a treat. Like everything new, it can take some time to understand it. We’ll just dive right in.
One of the deepest aspects of projective geometry is the principle of duality, whereby every figure or theorem has a dual partner. One of the most famous pair of dual theorems Pascal’s Theorem (1640), and *Brianchon’s Theorem (1814). We present the theorems using two columns, a traditional feature of writings on projective geometry. The reader can confirm that the theorems differ only by a small subset of terms, that will be discussed in more detail later.
Pascal’s Theorem #
The meeting points of opposite sides of a hexagon inscribed in a conic section lie on a common line.
Discussion #
The hexagon is given in the figure by $ABCA'B'C'$. The six sides are $\{AB,BC,CA',A'B',B'C',C'A\}$. The intersections of opposite sides are $A'' = BC \wedge B'C' $, etc. These three points always lie on a line!
Brianchon’s Theorem #
The joining lines of opposite vertices of a hexalateral circumscribed around a conic section are co-punctual.
Discussion #
The hexalateral is given in the figure by $abca'b'c'$. The six vertices are the intersection points $\{ab, bc, ca', a'b', b'c', c'a\}$. The joining lines of opposite vertices are $a'' = bc \vee b'c'$, etc. These three lines have a common point.