Proof of the Hexagon Theorem
The Hexagon Theorem is that, when the vertices of a hexagon lie on a conic section, and the point of intersection of each of the three pairs of opposite sides is taken, then the three intersection points lie on one straight line. Pappus proved the case when the “conic section” is a pair of straight lines; Pascal, a circle. By reviewing the work of Pappus, along with the use of it by Hessenberg to prove Desargues’s Theorem, we find a way to give a purely geometric definition of proportion, not using such an “Archimedean” assumption that Euclid does (in the definition attributed to Eudoxus).
Polytropy 