Recall the “fish-tail” diagram:
| \ / |
x | * | y z = x<+>y
| /|\ |
If you flip this diagram left for right, then the lengths x, y and z remain the same; so x<+>y = y<+>x. Commutativity is illustrated by reflection.
If you stretch it vertically then x, y and z become kx, ky and kz; so k(x<+>y) = kx <+> ky. So distribution is illustrated by stretching.
So commutativity and distribution for reduction relate to the symmetries of the fish-tail diagram. The associative law is more complicated. It is illustrated by this diagram:
AC, DE, FG, HI and JK are parallel;
ABC, CIJ, BGJ, BEH, AEGIK and ADFHJ are collinear.
If AC = x, JK = y and AB = z, then
IH = x<+>y ; GF = y<+>z ; DE = (x<+>y) <+>z
But (x<+>y)<+>z = x<+>(y<+>z);
So the AC and FG fishtail meets at E;
Therefore C, E and F are collinear.
If AC, DE, FG, IH and JK are parallel;
and ABC, CIJ, BGJ, BEH, AEGIK and ADFHJ are collinear;
then CEF are collinear.
That theorem in affine geometry is equivalent to the associative law for reduction. I therefore call this the Affine Reductive Associative Theorem. Here is a projective version of the diagram, with D omitted, and including L as a point at infinity:
Given these collinearities:
ABCL, FGL, HIL, JKL, CIJ, BGJ, BEH, AEGIK, AFHJ
Then CEF are collinear.
This theorem is a consequence of Pappus’s Hexagon Theorem: for BCL are collinear points and FHJ are collinear points; therefore the opposite sides of the hexagon BHLFCJ meet in three collinear points: EGI. This proof from Pappus does not assume that the three lines BCL, FHJ and EGI meet at a point, as they do here.
So the associativity of reduction is a consequence of a special case of Pappus’s Hexagon Theorem!