Intuitive explanation of Pascal's Theorem

Question

I am wondering why Pascal's Theorem, as well as other 'Euclidean' results in projective geometry like Brianchon's Theorem should be true for not only circles, but also conics in general. Is there some sort of projective transformation that sends conics to circles, and how exactly does projective geometry present such a nice setting for conics in the Euclidean plane?

Answer 1

In projective geometry, a circle is just a special case of a conic, namely one passing through the ideal circle points $(1,\pm i,0)$. So if you allow for complex transformations, then you can take a pair of conics and map two points of intersection to these special points to obtain a configuration with two circles. For more than two conics, though, this only holds true if all of the conics have two points of intersection (real or complex) in common.

A circle is a fundamentally Euclidean concept. Or perhaps I should say a metric-dependent object. In neutral projective geometry, with no points designated to play the role of the ideal circle points, you can't speak about circles but you can still speak about conics. For this reason, conics are the more natural objects in projective geometry. You will notice how a projective transformation tends to transform a circle to some non-circular conic in general.

The Euclidean plane is essentially the projective plane with the ideal circle points as distinguished points (and derived from them the line at infnity as a distinguished line). In the context of Cayley-Klein geometry one can distinguish any conic to introduce a metric on the projective plane. The pair of ideal circle points is a special degenerated conic, namely one where the primal representation describes the set of all points at infinity, as a line with multiplicity two, while the dual representation describes as tangents the set of all lines passing through either of these ideal circle points. But perhaps this is going way further than what you were actually asking.