Pappos’s Theorem: Nine Proofs and Three Variations

Πάππος ο Αλεξανδρεύς/ Pappus’s Theorem. free e book

We will begin our journey through projective geometry in a slightly uncommon way. We will have a very close look at one particular geometric theorem— namely The hexagon theorem of Pappos. Pappos of Alexandria lived around 290–350 CE and was one of the last great Greek geometers of antiquity.
He was the author of several books (some of them are unfortunately lost) that covered large parts of the mathematics known at that time. Among other topics, his work addressed questions in mechanics, dealt with the volume/ circumference properties of circles, and even gave a solution to the angle trisection problem (with the additional help of a conic). The reader may take this first chapter as a kind of overture to the remainder of the book in which several topics that are important later on are introduced. Without any harm
one can also skip this chapter on first reading and come back to it later…

1.1 Pappos’s Theorem and Projective Geometry
The theorem that we will investigate here is known as Pappos’s hexagon theorem and usually attributed to Pappos of Alexandria (though it is not clear whether he was the first mathematician who knew about this theorem).
We will later see that this theorem is special in several respects. Perhaps the most important property is that in a certain sense Pappos’s theorem is the smallest theorem expressible in elementary terms only. The only objects involved in the statement of Pappos’s theorem are points and lines, and the only relation needed in the formulation of the theorem is incidence. Properly
stated, the theorem consists only of nine points and nine lines, and there is no such theorem with fewer items. free e book