Proposition: The definition of A + C is independent of the choice of B.
We can now continue proving properties of addition in Desarguesian affine planes.
Proposition: For each point A on l, there is a point (-A) on l such that A + (-A) = (-A) + A = O.
This proposition shows that every point on l has an additive inverse.
We may also prove that addition is associative.
Proposition: Given points A, C and E on l, (A+C)+E = A + (C+E).
The final property that we need to prove is that addition is commutative. This proof uses the converse of Desargues theorem, so we need to know that the converse holds in any Desarguesian affine plane.
Theorem: In any Desarguesian projective plane, if two triangles are perspective from a line, then they are perspective from a point.
Pf: Let two triangles PQR and P'Q'R', be perspective from a line l. In other words, let l contain three points D,E,F such that D lies on both QR and Q'R', E on both RP and R'P' and F on both PQ and P'Q'. Consider the triangles PP'E and QQ'D. The point F is on PQ, P'Q' and ED, so these triangles are perspective from a point. By Desargues theorem, these triangles are also perspective from a line, meaning that PP' and QQ' meet at a point O, PE and QD meet at R, and P'E and Q'D meet at R' with O, R and R' collinear. Thus, PP', QQ' and RR' all meet at O and the original two triangles are perspective from a point.
We can now state an affine version of this converse, in the case that the line at infinity is the line that the two triangles are perspective from.
Affine Form of the Converse of Desargues Theorem: In a Desarguesian affine plane, if the three pairs of corresponding sides of two triangles are parallel, then the lines joining corresponding vertices of the triangles are either mutually parallel or meet at a common point.