proof of Pappus’s theorem

Pappus’s theorem says that if the six vertices of a hexagonMathworldPlanetmath lie alternately on two lines, then the three points of intersection of opposite sides are collinearMathworldPlanetmath. In the figure, the given lines are A11A13 and A31A33, but we have omitted the letter A.

The appearance of the diagram will depend on the order in which the given points appear on the two lines; two possibilities are shown.

Pappus’s theorem is true in the affine planeMathworldPlanetmath over any (commutative) field. A tidy proof is available with the aid of homogeneous coordinatesMathworldPlanetmath.

No three of the four points A11, A21, A31, and A13 are collinear, and therefore we can choose homogeneous coordinates such that

A11=(1,0,0)  A21=(0,1,0)
A31=(0,0,1)  A13=(1,1,1)

That gives us equations for three of the lines in the figure:

A13A11:y=z  A13A21:z=x  A13A31:x=y.

These lines contain A12, A32, and A22 respectively, so

A12=(p,1,1)  A32=(1,q,1)  A22=(1,1,r)

for some scalars p,q,r. So, we get equations for six more lines:

A31A32:y=qx  A11A22:z=ry  A12A21:x=pz (1)
A31A12:x=py  A11A32:y=qz  A21A22:z=rx (2)

By hypothesis, the three lines (1) are concurrentMathworldPlanetmath, and therefore prq=1. But that implies pqr=1, and therefore the three lines (2) are concurrent, QED.

Title proof of Pappus’s theorem
Canonical name ProofOfPappussTheorem
Date of creation 2013-03-22 13:47:40
Last modified on 2013-03-22 13:47:40
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Proof
Classification msc 51A05