Let three points A, B, C be incident to a single straight line and another three points a,b,c incident to* another straight line. Then three pairwise intersections 1 = BcbC, 2 = AcaC, and 3 = AbaB are incident to a (third) straight line.
*To make the dual terminology of a point lying on a line vs. a line passing through a point more nearly parallel, some authorities use the expression "incident to" to mean either one. For example, three collinear points are said to be incident to a particular line, and three concurrent lines are said to be incident to a particular point.
You can think of Pappus' Theorem as a special case of Pascal's Theorem, in which a hexagon is inscribed in a conic. The blue lines in the diagram are a special case of a conic, and the hexagon whose sides are AbCaBc is inscribed in this conic. Thus, the intersections of its opposite sides, Ab and aB; bC and Bc; and Ca and cA, are collinear.
. . . . . . proof needed.
Cut-the-knot: Pappus' Theorem gives a proof and a Java applet that lets you explore it.
Summary of geometrical theorems
Pappus-related things you might have been looking for when you found this page.
Pappus' Chain of Circles -- circles inscribed in an arbelos
Pappus' Centroid Theorem in solid geometry
The webmaster and author of this Math Help site is Graeme McRae.