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

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1 Mathematical Modelling and Applications 2018; 3(1: doi: /jmma ISSN: (Print; ISSN: (Online Three Vertex and Parallelograms in the ine Plane: Similarity and Addition Abelian Groups of Similarly n-vertexes in the Desargues ine Plane Orgest Zaka Department of Mathematics, Faculty of Technical Science, University Ismail QEMALI of Vlora, Vlora, Albania address: To cite this article: Orgest Zaka Three Vertex and Parallelograms in the ine Plane: Similarity and Addition Abelian Groups of Similarly n-vertexes in the Desargues ine Plane Mathematical Modelling and Applications Vol 3, No 1, 2018, pp 9-15 doi: /jmma Received: May 14, 2017; Accepted: December 18, 2017; Published: January 8, 2018 Abstract: 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 n- vertexes 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 ine Plane, Similarity of n-vertexes, Abelian Group 1 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] 2 n-vertexes in ine Plane and Their Similarity 21 3-Vertexes and Their Similarity Let's have the affine plane A = ( PLI,, Definition 211 Three-Vertex will called an ordered trio of non-collinear points (A,B,C in an affine plane Definition 212 Two three-vertexes (A 1,B 1,C 1 and (A 2,B 2,C 2 will call similar if they meet conditions: A 1 B 1 //A 2 B 2 ; A 1 C 1 //A 2 C 2 and B 1 C 1 //B 2 C 2 Example 211 In affine plane of the second order have the similar three-vertices (Figure 1: ( A D C ( B C D,,,, : because, ADBC; DCCD; ACBD ( A B D ( C D B,,,, : because, ABCD; BDDB; ADCB

2 10 Orgest Zaka: Three Vertex and Parallelograms in the ine Plane: Similarity and Addition Abelian Groups of Similarly n-vertexes in the Desargues ine Plane ( A B D ( D C A,,,, : because, ABDC; BDCA; ADDA since the parallelism in the affine plane is equivalence relation then will have to: A 1 B 1 //A 2 B 2 and A 2 B 2 //A 3 B 3 A 1 B 1 //A 3 B 3 ; A 1 C 1 //A 2 C 2 and A 2 C 2 //A 3 C 3 A 1 C 1 //A 3 C 3 ; B 1 C 1 //B 2 C 2 and B 2 C 2 //B 3 C 3 B 1 C 1 //B 3 C 3 Well, (A 1,B 1,C 1 (A 3,B 3,C Vertexes Definition 221: In affine planea, a set of four-point three out of three not-collineary will call 4-vertexes Definition 222: Two 4-vertexes ABCD and A B C D will call similar only if have the following parallels: AB A B, BC B C, CD C D and DA D A Figure 1 The similar three-vertexes in affine plane of order 2 Example 212: In the third order affine plane ( 2,3,9 ( 7,9,3, because: l l l l l l ; ; ; (2,3 (7,9 (2,9 (7,3 (3,9 (9,3 23 Parallelograms Definition 213: Parallelogram will call the ordered quartet of points (A,B,C,D fromp, that meets the conditions: AB//CD and BC//AD the lines AC and BD are called the diagonal of parallelogram Example 221: In affine plane of the second order (Figure 3 a have the following parallelogram: (A,D,B,C with the diagonal AB and DC (Figure 3 b; (A,B,D,C with the diagonal AD and BC (Figure 3 c; (A,B,C,D with the diagonal AC and BD (Figure 3 d Figure 2 Two similar three-vertexes in affine plane of order 3 Proposition 211: The similarity of the three-vertexes is equivalence relation Proof: 1 It is clear that every three-vertexes (A,B,C is similar to yourself ( A, B, C ( A, B, C 2 If three-vertexes (A 1,B 1,C 1 (A 2,B 2,C 2, are similar then also three-vertexes (A 2,B 2,C 2 (A 1,B 1,C 1, are similarity since from: A 1 B 1 //A 2 B 2 ; A 1 C 1 //A 2 C 2 ; B 1 C 1 //B 2 C 2 A 2 B 2 //A 1 B 1 ; A 2 C 2 //A 1 C 1 ; B 2 C 2 //B 1 C 1 3 If (A 1,B 1,C 1 (A 2,B 2,C 2, and three-vertexes (A 2,B 2,C 2 (A 3,B 3,C 3 then have to (A 1,B 1,C 1 (A 3,B 3,C 3, because parallelism in the affine plane is equivalence relation, which is described in [2], [3], [4] So would have to: A 1 B 1 //A 2 B 2 ; A 1 C 1 //A 2 C 2 ; B 1 C 1 //B 2 C 2 and A 2 B 2 //A 3 B 3 ; A 2 C 2 //A 3 C 3 ; B 2 C 2 //B 3 C 3 Figure 3 4-parallelograms in the affine plane of order 2 From the definition of parallelogram and the fact that parallelism is the equivalence relation is evident this Proposition: Proposition 221: If you have two similar 4-vertexes, where each is parallelogram then another 4-vertexes will be parallelogram 24 n-vertexes Definition 241: In affine planea, a set of n-points non-

3 Mathematical Modelling and Applications 2018; 3(1: collinearly three out of three will call n-vertex Definition 242: Two n-vertexes (A 1 A 2 A n and (B 1 B 2 B n will call similar just if have the following parallelisms: A i A j B i B j, (i,j {(1,1,,(1,n;(2,1,,(2,n; ;(n,1,,(n,n} 3 The Addition of Similarity Three- Vertexes in the Desargues ine Plane Let's have two similarity three-vertexes (A 1,A 2,A 3 and A = PLI,, (B 1,B 2,B 3 in the Desargues affine plane ( Constructed the lines A 1 B 1, A 2 B 2, A 3 B 3, since are in Desargues affine plane and the similarity of three-vertexes have to: A 1 A 2 B 1 B 2 ; A 2 A 3 B 2 B 3 ; A 1 A 3 B 1 B 3 the lines A 1 B 1, A 2 B 2 and A 3 B 3, or will be parallel or will cross the on a single point Receive now a point O 1 A 1 B 1, and find points D O 2 and O 3 how: O AA O =AB l and O = AB l O AA So have obtained thus three-vertexes (O 1,O 2,O 3, (points O 1, O 2 and O 3 are non-collinearly, because from construction this three-vertexes will be similar with three-vertexes (A 1,A 2,A 3 where O 1 A 1 B 1, O 2 A 2 B 2 and O 3 A 3 B 3 This three-vertex called zero three-vertex So have three lines, to which each have its zero point Now just as to [3], additions of the points of each line based on the algorithm of additions of points in a line in Desargues affine plans, and take: C 1 =A 1 +B 1, C 2 =A 2 +B 2, C 3 =A 3 +B 3 (1 Definition 31: The addition of two similarity threevertexes (A 1,A 2,A 3 and (B 1,B 2,B 3,called three-vertexes (C 1,C 2,C 3, where the points (vertexes C 1,C 2,C 3 They found according to equation (1 (Figure 4 Figure 4 The Addition of two similarity three-vertexes in the Desargues ine Plane From construction of three-vertexes as the addition of two similar three-vertexes have evident this Proposition: Proposition 31 Three-vertexes that obtained as the sum of two similar three-vertexes (A 1,A 2,A 3 and (B 1,B 2,B 3, it is similar to the first two Well and (A 1 + B 1, A 2 +B 2, A 3 +B 3 (A 1,A 2,A 3 (A 1 + B 1, A 2 +B 2, A 3 +B 3 (B 1,B 2,B 3 Proposition 32 The additions of non-similarity threevertexes it may not be a three-vertexes Proof: If renounce above from addition algorithm of the similar three-vertexes In the same logic, are additions together two of whatever three-vertexes Let's have two whatever three-vertexes (A 1,A 2,A 3 and (B 1,B 2,B 3 in the affine plane = ( A PLI,, Construct the line A 1 B 1, A 2 B 2 and A 3 B 3 Get a whatever three-vertexes (O 1,O 2,O 3 (the points O 1, O 2 and O 3 are non-collinearly where O 1 A 1 B 1, O 2 A 2 B 2 and O 3 A 3 B 3 This three-vertexes called the zero three-vertex So have three lines, where, in each line have hers zero point Now just as to [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

4 12 Orgest Zaka: Three Vertex and Parallelograms in the ine Plane: Similarity and Addition Abelian Groups of Similarly n-vertexes in the Desargues ine Plane Figure 5 The Addition of two non-similarity three-vertexes in the Desargues ine Plane is a three-vertex Defined in this way, it seems as if does not have a contradiction But the veracity of this Proposition are presenting with the help of a simple anti-example, shown in the following figure Figure 6 The Addition of two non-similarity three-vertexes in the Desargues ine Plane is not a three-vertex Remark 31: 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 Remark 32: Marked the set of three-vertexes in the Desargues affine plans with symbol T Remark 33: Marked the set of similarity three-vertexes in the Desargues affine plans with symbolt D D It is clear that: T T Let us be ( A,A,A and ( B,B,B two whatsoever three-vertexes in the sett I associate pairs D D ( A,A,A,B,B,B ( T T, D three-vertex ( C,C,C T, that the vertexes are determines with algorithm in [3] According to the preceding C,C,C is determined in Theorems, three-vertexes ( single mode by [3] Thus obtain an application D D D T T T Definition 32: In the above conditions, application defined by + T D T D T D :,

5 Mathematical Modelling and Applications 2018; 3(1: ( A,A,A,B,B,B ( ( C,C,C D D ( A,A,A,B,B,B ( T T The addition in T write according to this Definitions, can D ( A,A,A,B,B,B ( T, 1 P1 AB,AB,AB, P1 A1 2 l AB l 1 1 OP=P, P2 3 l PB AB 1 1=C P 1 A2 4 lab l 2 2 OP=P, P 3 5 lpb AB 2 2=C P1 A3 6 lab l OP=P, 3 1 P 4 7 lpb AB 3 3=C = ( A,A,A + ( B,B,B ( C,C,C Theorem 31: For every two three-vertexes ( A,A,A, D ( B,B,B T, algorithm (2 determines the single D three-vertexes ( C,C,C T depend on the choice of hers auxiliary point P 1 (2, which does not Proof: From Theorem 21, in [3], have to addition of two points in a line of Desargues affine plane does not depend on the choice of hers auxiliary point For this reason keep as auxiliary points for addition of pairs points, the auxiliary point P 1 From Theorem 31, appears immediately true this Proposition 33: Additions of three-vertexes int there are element zero the three-vertexes ( O,O,O : D ( A,A,A T, ( A,A,A + ( O,O,O = ( O,O,O ( A,A,A = ( A,A,A = + = As well as worth and below Propositions Proposition 34: Additions of three-vertexes is commutative int : ( A,A,A,B,B,B ( ( A,A,A + ( B,B,B = = ( B,B,B + ( A,A,A T D (3 (4 have: ( A,A,A + ( B,B,B = ( A + B,A + B,A + B From Theorem 21, 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 [1] ( B A,B A,B A ( B,B,B ( A,A,A = = Proposition 35: Addition of three-vertexes is associative int : ( A,A,A,B,B,B (,( C,C,C T ( A,A,A + ( B,B,B + ( C,C,C = = ( A,A,A + ( B,B,B + ( C,C,C Proof: Let's have three whatever three-vertexes D ( A,A,A,B,B,B ( ( C,C,C T, Appreciate now, D ( A,A,A + ( B,B,B + ( C,C 1 2,C 3 = = ( A,A,A + ( B+C,B C 2,B3 + C3 = A + ( B+C,A + ( B +C, A + ( B + C [1] ( (,( ( A1 B1, A2 B,A 2 3 B3 ( C,C 1 2,C 3 ( A,A,A + ( B,B,B ( C,C,C = A + B +C,A + B +C A + B + C = = + Proposition 36: For every three-vertex in T exists her right symmetrical according to addition: D ( A,A,A ( A,A,A D T T,, ( ( = ( (5 A,A,A + A,A,A O,O,O (6 D Proof: Let us have whatever ( A,A,A T, fix the zero three-vertexes ( O,O,O T (which would be similar to three-vertexes ( A,A,A if apply the Proposition 3 4, in [3] pp34990, have that, for points A 1, A 2 and A 3, find points respectivelya 1 OA,A OA 2 2 and A3 OA 3 3 such that: A+A = O ; A+A = O;A + A = O D A,A,A = A,A,A T such that it: Well ( ( D Proof: By definition of additions of three-vertexes that

6 14 Orgest Zaka: Three Vertex and Parallelograms in the ine Plane: Similarity and Addition Abelian Groups of Similarly n-vertexes in the Desargues ine Plane ( A,A,A + ( A,A,A = ( O,O,O I summarize what was said earlier in this Theorem 32: The Groupoid (, + (abelian Group T is commutative 4 The Addition of Similarity n-vertexes in the Desargues ine Plane Equally as addition of three-vertexes in Desargues affine plane, by the same logic, additions and n-vertexes in this plane Remark 41: The set of similarity n-vertexes in Desargues affine plane marked with symbol N The addition algorithm of n-vertexes, by analogy with addition algorithm of the three-vertexes are presenting below: Let's have two whatever similarity n-vertexes in Desargues affine plane: D ( A,A,A A,B,B,B ( B N,,,, n n The definitions of the similarity of n-vertexes have the following parallelisms: AA BB, AA BB,, A A B B, AABB n 1 n n 1 n n 1 n 1 Constructed the linesab,ab,ab ,, AB n n, 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 pointo1 AB 1 1, and then find all the other vertexes of n-vertexes how: O1 O1 O AA AA n n n 1 3 AA 1 n O =AB l, O = AB l,, O = AB l Definition 41: In the above conditions, application defined by D D D + : N N N, ( A,A,, A,( B,B,, B ( C,C,, C 1 2 n 1 2 n 1 2 n D ( A,A,A A,B,B,B ( B n n,,,, N call the addition in N according to this Definitioni, can write the addition algorithm of the n-vertexes: D ( A,A,A,, A,B,B,B (,, B N n n 1 P AB,AB,,AB, n n ( P1 A1 i lab l 1 1 OP =P, P2 ( ii l PB AB 1 1=C1 1 1 P1 A2 ( i lab l 2 2 OP=P, P3 ( ii l PB AB 2 2=C2 1 2 P1 A3 ( i lab l OP=P, P4 ( ii l PB AB 3 3=C ( i n+1 ( l P1 An AB l n n OnP1 P =P, n+1 n+1 ii l PB AB n n=c n 1 n ( A,A, A + ( B,B, B ( C,C, C,, =, 1 2 n 1 2 n 1 2 n And for n-vertexes, have true analog the statements had to three-vertexes (everything proved equally Well have the verities of following statements Theorem 41: For every two n-vertexes ( A,A, 1 2, A n, D ( B,B, B N, algorithm (7 determines the 1 2, n D 1 2, n N, which does not depend on the choice of hers auxiliary point P 1 single three-vertexes ( C,C, C From Theorem 41, appears immediately true this Proposition 41: Additions of n-vertexes in there are element zero the three-vertexes ( O,O, 1 2, O n : D ( A,A, 1 2, An N, ( O,O, 1 2, On ( A,A, 1 2, An + ( O,O, 1 2, On = ( O,O, 1 2, On + ( A,A, 1 2, An = ( A,A,, A N (7 D N 1 2 Also well as worth and below Propositions n Proposition 42: Additions of n-vertexes is commutative in N : D ( A,A,, A,B,B, (, B N 1 2 n 1 2 n ( A,A, 1 2, A n + ( B,B, 1 2, Bn = ( B,B,, B + ( A,A,, A 1 2 n 1 2 Proposition 43: Addition of n-vertexes is associative in N : n D ( A,A,, A,B,B, (, B,( C,C,, C N 1 2 n 1 2 n 1 2 n ( A,A, 1 2, A n + ( B,B, 1 2, Bn + ( C,C 1 2,, Cn = = ( ( + ( (8 A,A, 1 2, A n + B,B, 1 2, Bn C,C 1 2,, C n (10 (9

7 Mathematical Modelling and Applications 2018; 3(1: Proposition 44: For every n-vertex in N exists her right symmetrical according to addition: D ( A,A,, A N, ( A,A,, A D 1 2 n 1 2 n N (, (, = (, A,A, A + A,A, A O,O, O ( n 1 2 n 1 2 n (Here have that ( 1 2, n = ( 1 2, n A,A, A A,A, A By Theorem 41, Propositions 41, 42, 43 and 44 we have this true theorem: Theorem 42: The Groupoid ( N, + commutative (Abelian Group References [1] Dr Orgest ZAKA, Prof Dr Kristaq FILIPI, (2017 "An Application of Finite ine Plane of Order n, in an Experiment Planning", International Journal of Science and Research (IJSR, Volume 6 Issue 6, June 2017, , DOI: /ART [2] Zaka, O, Flipi, K (2016 One construction of an affine plane over a corps Journal of Advances in Mathematics, Council for Innovative Research Volume 12 Number is [3] Zaka, O, Filipi, K (2016 The transform of a line of Desargues affine plane in an additive group of its points, International Journal of Current Research, 8, (07, [4] FRANCIS BORCEUX (2014 An Axiomatic Approach to Geometry&An Algebraic Approach to Geometry (Geometric Trilogy I& Geometric Trilogy II Springer International Publishing Switzerland [5] GRRILLET, P A (2007 Abstract Algebra(Second Edition Graduate Texts in Mathematics v242 ISBN-13: Springer Science + Business Media, LLC (Grrillet, P A, et al, 2007 [6] HILBERT D, VOSSEN S C (1990 Geometry And The Imagination Chelsea Publishing Company [7] Buenkenhout, F, (Editors(1995 HANDBOOK OF INCIDENCE GEOMETRY Elsevier Science B V ISBN: X [8] Hungerford, TH W (1974 Algebra (Graduate Text in Mathematics vol 73 Springer-Verlag New York, Inc ISBN [9] Lang, S (2002 Abstract Algebra (Third Edition Springerverlag new york, inc ISBN X [10] Ueberberg, Johannes (2011, Foundations of Incidence Geometry, Springer Monographs in Mathematics, Springer, doi:101007/ , ISBN