In mathematics,
Pappus's hexagon theorem (attributed to
Pappus of Alexandria) states that given one set of
collinear points
A,
B,
C, and another set of collinear points
a,
b,
c, then the intersection points
X,
Y,
Z of
line pairs
Ab and
aB,
Ac and
aC,
Bc and
bC are
collinear, lying on the
Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon
AbCaBc. It holds in a
projective plane over any field, but fails for projective planes over any noncommutative
division ring. Projective planes in which the "theorem" is valid are called
pappian planes.