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 elsewhere on this site.
Pascal’s Theorem #
<img src="/images/projGeomThms/pascalThm.png"
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The meeting points of opposite sides of a hexagon inscribed in a conic section are collinear.
Discussion #
The hexagon is given in the figure by $ABCA'B'C'$. The six sides are $\{AB,BC,CA',A'B',B'C',C'A\}$. Their intersections of opposite sides are $A'' = BC \wedge B'C' $, etc. These three points always lie on a line!
Brianchon’s Theorem #
<img src="/organic-geometry/images/projGeomThms/brianchonThm.png"
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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 $\{ab, bc, ca', a'b', b'c', c'a\}$. Their joining lines are $a'' = bc \wedge b'c'$, etc. These three lines always have a common point!