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

 * PappusHexagonTheorem.png

This was first proved by him in about 300 CE.

The theorem is stated as Propositions $138$, $139$, $141$, and $143$ of Book VII of 's Synagoge.