# Pappus's Hexagon Theorem

## Theorem

Let $A, B, C$ be a set of collinear points.

Let $a, b, c$ be another set of collinear points.

Let $X, Y, Z$ be the points of intersection of each of the straight lines $Ab$ and $aB$, $Ac$ and $aC$, and $Bc$ and $bC$.

Then $X, Y, Z$ are collinear points.

## Also known as

This theorem is also known just as Pappus's Theorem.

## Source of Name

This entry was named for Pappus of Alexandria.

## Historical Note

Pappus's Hexagon Theorem was first proved by Pappus of Alexandria in about $300$ CE.

The theorem is stated as Propositions $138$, $139$, $141$, and $143$ of Book $\text{VII}$ of Pappus's Collection.

It is noted that it is a limiting case of Pascal's Mystic Hexagram.

In $1899$ its full significance was revealed by David Hilbert, during his work on clarifying the foundations of geometry.