Pappus's Hexagon Theorem

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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

PappusHexagonTheorem.png



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