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[parent] proof of Pappus's theorem (Proof)

Pappus's theorem says that if the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of opposite sides are collinear. In the figure, the given lines are $ A_{11}A_{13}$ and $ A_{31}A_{33}$, but we have omitted the letter $ A$.

\includegraphics{pappus}

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 plane over any (commutative) field. A tidy proof is available with the aid of homogeneous coordinates.

No three of the four points $ A_{11}$, $ A_{21}$, $ A_{31}$, and $ A_{13}$ are collinear, and therefore we can choose homogeneous coordinates such that

$\displaystyle A_{11}=(1,0,0)\qquad A_{21}=(0,1,0)$
$\displaystyle A_{31}=(0,0,1)\qquad A_{13}=(1,1,1)$
That gives us equations for three of the lines in the figure:
$\displaystyle A_{13}A_{11}:y=z\qquad A_{13}A_{21}:z=x\qquad A_{13}A_{31}:x=y\;.$
These lines contain $ A_{12}$, $ A_{32}$, and $ A_{22}$ respectively, so
$\displaystyle A_{12}=(p,1,1)\qquad A_{32}=(1,q,1)\qquad A_{22}=(1,1,r)$
for some scalars $ p,q,r$. So, we get equations for six more lines:
$\displaystyle A_{31}A_{32}:y=qx\qquad A_{11}A_{22}:z=ry\qquad A_{12}A_{21}:x=pz$ (1)

$\displaystyle A_{31}A_{12}:x=py\qquad A_{11}A_{32}:y=qz\qquad A_{21}A_{22}:z=rx$ (2)

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



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Cross-references: QED, implies, concurrent, hypothesis, scalars, equations, homogeneous coordinates, field, commutative, affine plane, collinear, opposite sides, points, lines, hexagon, Pappus's theorem

This is version 3 of proof of Pappus's theorem, born on 2003-07-26, modified 2003-08-10.
Object id is 4509, canonical name is ProofOfPappussTheorem.
Accessed 4810 times total.

Classification:
AMS MSC: 51A05 (Geometry :: Linear incidence geometry :: General theory and projective geometries)
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