Three Vertex and Parallelograms in the Affine Plane: Similarity and Addition Abelian Groups of Similarly n-Vertexes in the Desargues Affine Plane

In this article will do a’ concept generalization n-gon. By renouncing the metrics in much axiomatic geometry, the need arises for a new label to this concept. In this paper will use the meaning of n-vertexes. As you know in affine and projective plane simply set of points, blocks and incidence relation, which is argued in [1], [2], [3]. In this paper will focus on affine plane. Will describe the meaning of the similarity n-vertexes. Will determine the addition of similar three-vertexes in Desargues affine plane, which is argued in [1], [2], [3], and show that this set of three-vertexes forms an commutative group associated with additions of three-vertexes. At the end of this paperare making a generalization of the meeting of similarity nvertexes in Desargues affine plane, also here it turns out to have a commutative group, associated with additions of similarity n-vertexes. Keyword: n-vertexes, Desargues Affine Plane, Similarity of n-Vertexes, Abelian Group


Introduction
In Euclidian geometry use the term three-angle and non three-vertex, this because the fact that the Euclidean geometry think of associated with metrics, which are argued in [4], [6], [7]. In this paper will use the "three-Vertex" term, by renouncing the metric. Will generalize so its own meaning in the Euclidean case. With the help of parallelism [1], [2], [3] will give meaning of similarity and will see that have a generalization of the similarity of the figures in the Euclidean plane. By following the logic of additions of points in a line of Desargues affine plane submitted to [3], herewill show that analogously this meaning may also extend to the addition of similarity three-vertex in Desargues affine plane, moreover extend this concept for the similarity n-vertexes to the Desargues affine plane.
The aim is to see if the move to three-vertexes as well as to n-vertexes has the group's properties, which are arguing that the best in [5], [8], [9].
So would have to:  3 C 3 ; B 2 C 2 //B 3 C 3 since the parallelism in the affine plane is equivalence relation then will have to: AB||A'B', BC||B'C', CD||C'D' and DA||D'A'.  From the definition of parallelogram and the fact that parallelism is the equivalence relation is evident this Proposition: Proposition 2.2.1: If you have two similar 4-vertexes, where each is parallelogram then another 4-vertexes will be parallelogram.

The Addition of Similarity Three-Vertexes in the Desargues Affine Plane
(1)  From construction of three-vertexes as the addition of two similar three-vertexes have evident this Proposition:  [3], the addition points of every line based on the addition algorithm given to [3], and take: C 1 =A 1 +B 1 , C 2 =A 2 +B 2 and C 3 =A 3 +B 3 . Similarly n-Vertexes in the Desargues Affine Plane  Remark 3.1: By following the addition algorithms for points in a line of Desargues affine plane, is sufficient to get only an auxiliary point P 1 , for this obedient from [3], for three sums can either take one three-vertexes (P 1 ,P 2 ,P 3 ), wherein each point of three-vertexes be auxiliary point for the relevant sum.  A ,A , B ,B , that the vertexes are determines with algorithm in [3]. According to the preceding Theorems, three-vertexes ( )  A ,A ,A , B ,B , A ,A ,A , B ,B A ,A ,A , B ,B P A B ,A B ,A B ,

A ,A ,A A ,A ,A O ,O ,O O ,O ,O A ,A ,A A ,A ,A
As well as worth and below Propositions.  A ,A , B ,B ,B   A ,A ,A + B ,B ,B   B ,B , A ,A ,A + B ,B ,B  A B ,A B ,A B From Theorem 2.1, in [3], have that for every two points is a line in the Desargues affine plane the addition is commutative, and consequently have to:  A ,A ,A + B ,B ,B  A B ,A B ,A B   B A ,B A ,B A  B ,B ,B  A ,A ,A A ,A , B ,B ,B C ,C ,C   A ,A ,A + B ,B , 2  3  1  2  3  1  2  3   1  2  3  1  1  2  2  3  3   1  1  1  2  2  2  3  3  3   [1]   1  1  1  2  2  2  3  3  3   1  1  2  2  3  3  1  2  3   1  2  3  1  2  3  1  2  3 , , ,

Proposition 3.5: Addition of three-vertexes is associative in
.

= A ,A ,A + A ,A ,A O ,O ,O
Proof: Let us have whatever ( )

= A ,A ,A + A ,A ,A O ,O ,O
I summarize what was said earlier in this

The Addition of Similarity n-Vertexes in the Desargues Affine Plane
Equally as addition of three-vertexes in Desargues affine plane, by the same logic, additions and n-vertexes in this plane.

A A B B A A B B A A B B A A B B
Constructed the lines 1 1 2 2 3 3 ,..., n n A B ,A B ,A B A B , since are in Desargues affine plane, and from the parallels the above, are the conditions of the Desargues theorem, it results that the above lines or crossing from a fixed point V or they have a bunch of parallel lines.
In both cases equally found the zero n-vertex. Take one first point 1   P A B ,A B ,...,A B And for n-vertexes, have true analog the statements had to three-vertexes (everything proved equally).
Well have the verities of following statements Also well as worth and below Propositions.