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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


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