### Pappus' Theorem

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.

### Internet references

Cut-the-knot:
Pappus' Theorem gives a proof and a Java applet that lets you
explore it.

### Related pages in this website:

Summary of geometrical theorems

Projective Geometry

**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.