Pascal's theorem

We may therefore now state Pascal’s and Brianchon’s theorem thus— Pascal’s Theorem.—If a hexagon be inscribed in a conic, then the intersections of opposite sides lie in a line.

Pascal’s Theorem.—If a solid angle of six faces be inscribed in a cone of the second order, then the intersections of opposite faces are three lines in a plane.

The applications of this theorem are very numerous; for instance, we derive from it Pascal’s theorem of the inscribed hexagon.

Also a conic can be defined as the locus of the points found by the usual construction, based upon Pascal’s theorem, for points on the conic through five given points.

It is convenient, in making use of Pascal’s theorem, to number the points, to indicate the order in which they are to be taken in forming a hexagon, which, by the way, may be done in 60 different ways.