# proof of Pappus’s theorem

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 $A$.

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

$${A}_{11}=(1,0,0)\mathit{\hspace{1em}\hspace{1em}}{A}_{21}=(0,1,0)$$ |

$${A}_{31}=(0,0,1)\mathit{\hspace{1em}\hspace{1em}}{A}_{13}=(1,1,1)$$ |

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

$${A}_{13}{A}_{11}:y=z\mathit{\hspace{1em}\hspace{1em}}{A}_{13}{A}_{21}:z=x\mathit{\hspace{1em}\hspace{1em}}{A}_{13}{A}_{31}:x=y.$$ |

These lines contain ${A}_{12}$, ${A}_{32}$, and ${A}_{22}$ respectively, so

$${A}_{12}=(p,1,1)\mathit{\hspace{1em}\hspace{1em}}{A}_{32}=(1,q,1)\mathit{\hspace{1em}\hspace{1em}}{A}_{22}=(1,1,r)$$ |

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

$${A}_{31}{A}_{32}:y=qx\mathit{\hspace{1em}\hspace{1em}}{A}_{11}{A}_{22}:z=ry\mathit{\hspace{1em}\hspace{1em}}{A}_{12}{A}_{21}:x=pz$$ | (1) |

$${A}_{31}{A}_{12}:x=py\mathit{\hspace{1em}\hspace{1em}}{A}_{11}{A}_{32}:y=qz\mathit{\hspace{1em}\hspace{1em}}{A}_{21}{A}_{22}:z=rx$$ | (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.

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 |