# 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.

## Proof

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## Also known as

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

## Also see

## 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.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{V}$: "Greatness and Misery of Man" - 1952: T. Ewan Faulkner:
*Projective Geometry*(2nd ed.) ... (previous) ... (next): Chapter $1$: Introduction: The Propositions of Incidence: $1.1$: Historical Note - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $9$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.8$: Pappus (fourth century A.D.) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $9$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Pappus' theorems**(3) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Pappus' theorems**(3)