Affine maps and some theorems in plane geometry

HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS NGUYEN THI THU AFFINE MAPS AND THREE THEOREMS IN PLANE GEOMETRY GRADUATION THESIS Major: Geometry Hanoi - 2019 HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS NGUYEN THI THU AFFINE MAPS AND THREE THEOREMS IN PLANE GEOMETRY GRADUATION THESIS Major: Geometry Supervisor: Assoc.Prof Nguyen Nang Tam Hanoi - 2019 Thesis Acknowledgements I would like to express my gratitudes to the teachers of the Department of Mathematics, Hanoi Pedagogical University 2, the teachers in the geometry group as well as the teachers involved The lecturers have imparted valuable knowledge and facilitated for me to complete the course and the thesis In particular, I would like to express my deep respect and gratitude to Assoc.Prof.Nguyen Nang Tam, who has direct guidance, help me complete this thesis Due to time, capacity and conditions are limited, so the thesis can not avoid errors Then, I look forward to receiving valuable comments from teachers and friends Hanoi, May - 2019 Student Nguyen Thi Thu Assurances I assure that the data and the results of this thesis are true and not identical to other topic I also assure that all the help for this thesis has been acknowledged and that the results presented in the thesis has been identified clearly Hanoi, May - 2019 Student Nguyen Thi Thu Contents Acknowledgements Thesis Assurance Introductions 1 AFFINE MAPS 1.1 1.2 1.3 Affine space 1.1.1 Definition of affine space 1.1.2 Examples 1.1.3 Some properties Affine maps 1.2.1 Definition 1.2.2 Examples 1.2.3 The properties of affine maps 1.2.4 Images and pre-image of flats by affine mapping 12 1.2.5 The single ratio and affine map 12 1.2.6 The parallel projections 14 Affine transformations 15 i Graduation thesis 1.4 Nguyen Thi Thu 1.3.1 Affine Isomorphism 15 1.3.2 Affine transformation 16 1.3.3 Affine mapping in affine coodinates 19 1.3.4 Homological affine mapping 20 1.3.5 Homologicalgline affine mapping 22 1.3.6 Involutory affine 24 Exercises 26 THREE THEOREMS IN PLANE GEOMETRY 32 2.1 The single ratio in plane 32 2.2 Thales’ theorem 33 2.3 Desargues’ theorem 35 2.3.1 Ceva’s theorem 35 2.3.2 Menelaus’ theorem 36 2.3.3 Desargues’ theorem 36 2.4 Pappus’ theorem 40 2.5 Exercises 42 Conclusions 45 References 45 ii Graduation thesis Nguyen Thi Thu Introductions Rationale Pure affine geometry, in the sense that there are no distance, angles, perpendicular Instead of, it is the notions of subtraction of points to produce a vector During the process, I learned about affine geometry and found that it has many applications and played an important role in mathematics With the desire to study more deeply in geometry and to be guided by the supervisor, I chose the topic Affine maps and some theorems in plane geometry for the graduation thesis Aims of the study The purpose of this thesis reseaches into affine mappings and three theorems of affine geometry in the plane The object and scope of the study The affine mappings and three theorems of affine geometry in the plane The references relate to affine geometry Research methods To research the textbooks, references,document relate to affine geometry Main contents The thesis consists of chapters Chapter 1: Affine maps 1.1 Affine spaces Graduation thesis Nguyen Thi Thu 1.2 Affine maps 1.3 Affinities Chapter 2: Three theorems in plane geometry 2.1 Simple Ratio 2.2 Thales theorem 2.3 Desargues theorem 2.4 Pappus theorem Chapter AFFINE MAPS In Chapter 1, we will present definition and some properties of affine space, affine maps and affinities The content is written based on references [1], [2], [3], [4], [5] 1.1 Affine space 1.1.1 Definition of affine space Definition 1.1.1 (see,[1- pp.1,2]) Let E be a K -vector space (where K is a field) An affine space over E is a set A together with a map A×E →A (M, v) → M + v = N such that: → − → − (1) M + = M for all M ∈ A, where is the identity element of E; Graduation thesis Nguyen Thi Thu − − − − − (2) M + (→ v +→ w ) = (M + → v ) + w for all M ∈ A and → v ,→ w ∈ E; and − (3) given M, N ∈ A, there exists a unique v ∈ E such that M + → v = N − Remark 1.1.2 (See,[1- p.2]) Note that if M ∈ A and → v ∈ E , the − − notation M +→ v means only the image of the pair (M, → v ) via the above map A × E → A Hence, the four signs + appearing in condition have different meanings: three of them represent the above map, and the other, ordinary vector addition in the vector space E Remark 1.1.3 (see,[1- page 2]) The unique vector determined by the −−→ points M and N is denoted by M N Hence, we have the fundamental −−→ relation M + M N = N Remark 1.1.4 The dimension of an affine space A is defined to be the dimension of its associated vector space E We shall write dimA = dimE 1.1.2 Examples Example 1.1.5 (see [1-page 3]) The standard example of an affine space is given by A = E, that is, the points of this affine space are the elements of the vector space The action is A×E →A − − (M, → v)→M +→ v where the sum is ordinary vector addition Chapter THREE THEOREMS IN PLANE GEOMETRY In Chapter, we will present the single ratio in plane, three theorem in plane : Thales’ theorem, Desargues’ theorem and Pappus’ theorem and give some exercise to apply three theorem The content is written base on references [2], [3], [6] 2.1 The single ratio in plane Definition 2.1.1 In A2 Let P,Q,R be three distinct collinear points −→ −→ Then we have a number λ ∈ R such that RP = λRQ The number λ is called the single ratio of triple point p,Q,R and denoted by λ = [P, Q, R] = (P, Q, R) Assume that R has character different from If [P, Q, R] = −1 then R is called midpoint of pair point (P,Q) Notation that the hypothesis P = Q then λ = 32 Graduation thesis Nguyen Thi Thu Figure 2.1: Thales’ theorem 2.2 Thales’ theorem Theorem 2.2.1 Let a, b and c be three distinct parallel lines, l1 and l2 two lines that are not parallel to a, b and c Let Ai = li ∩ a, Bi = li ∩ b, Ci = li ∩ c Then the equality −−−→ −−−→ A C1 A2 C2 −−−→ = −−−→ A1 B1 A2 B2 holds Conversely, if a point M of l1 satisfies the equality −−−→ −−−→ A1 M A2 C2 = −−−→ −−−→ A1 B1 A2 B2 then it belongs to c ( and M ≡ C1 ) Remark 2.2.2 If A,B,C and D are four collinear points ( with C = D), the ratio −−→ AB −−→ CD is well defined; this is the scalar λ such that 33 Graduation thesis Nguyen Thi Thu −→ −−→ AB = λCD Proof See Figure 2.1.Consider π : l1 → l2 is parallel projection − defined by a Then π is an affine map and → π is the associated linear map of π We have: π(A1 ) = A2 , π(B1 ) = B2 ,π(C1 ) = C2 −−−→ −−−→ −−−→ −−−→ − − Moreover, if A1 C1 = λA1 B1 , we get → π (A1 C1 ) = λ→ π (A1 B1 ) because −−−→ −−−→ → − π is linear This is to say A C = λA B , an equality from which the 2 2 direct sense of the theorem follows The coverse statement is a consequence of the direct one: we have −−−→ −−−→ A2 C2 −−−→ A1 C1 = −−−→ A1 B1 A2 B2 −−−→ −−−→ so thatA1 M = A1 C1 and thus M ≡ C1 ✷ Corollary 2.2.3 Let l1 and l2 be two lines intersecting at A, and a and b two parallel lines interseting li at Ai , Bi distict from A Then −−→ −−→ −−−→ AA2 A1 A2 AA1 −−→ = −−→ = −−−→ AB1 AB2 B1 B2 Proof See Figure 2.2 Draw a line parallel to a and b through A and apply Thales’ theorem, which gives the first equality The second ✷ equality follows 34 Graduation thesis Nguyen Thi Thu Figure 2.2: and one of its corollaries 2.3 2.3.1 Desargues’ theorem Ceva’s theorem Theorem 2.3.1 In A2 , let ABC be a triangle, and P, Q, R points of the sides BC, CA, AB The lines AP, BQ, CR are parallel or concurrent if and only if they satisy the equality [BCP ] [CAQ] [ABR] = −1 35 Graduation thesis Nguyen Thi Thu Figure 2.3: Ceva’s Theorem 2.3.2 Menelaus’ theorem Theorem 2.3.2 Let ABC be a triangle, and P, Q, R points of the side BC, CA, AB ( which has no point coincide with A, B, C ) Three points P, Q, R are collinear if and only if they satisfy the equality [P BC] [QCA] [RAB] = 2.3.3 Desargues’ theorem Theorem 2.3.3 Let ABC and A B C be two triangles Then the followinng statements are equality: a) Three lines AA , BB , and CC are concurrent b) The intersection of AB and A B , BC and B C , and AC and A C are collinear 36 Graduation thesis Nguyen Thi Thu Figure 2.4: Menelaus’ theorem Figure 2.5: Desargues’ theorem 37 Graduation thesis Nguyen Thi Thu Proof [ a) implies b) ] Assume that O is the intersection point of AA , BB and CC Let M, N, P be the intersection point of BC and B C , AB and A B , and CA and C A , respectively We need show that M, N, P are colinear Applying Menelaus’ theorem in three triangles OBC, OCA, OAB In the triangle OBC, three points M, B , C are collinear, then −−→ −−→ −−→ MB C C B O −−→ −−→ −−→ = MC C O B B (1) In the triangle OCA, three points N, A , C are collinear, then −−→ −−→ −−→ NC A A C O −−→ −−→ −−→ = NA A O C C (2) In the triangle OAB, three points P, A , B are collinear, then −→ −−→ −−→ PA B B A O −−→ −−→ −−→ = PB B O A A (3) Multiplication side to side of (1), (2), (3) we get: −−→ −−→ −→ MB NC P A −−→ −−→ −−→ = MC NA P B Consider the triangle ABC has M ∈ BC, N ∈ AC, P ∈ AB and −−→ −−→ −→ MB NC P A −−→ −−→ −−→ = MC NA P B 38 Graduation thesis Nguyen Thi Thu then, by Menelaus’ theorem M, N, P are collinear [ b) implies a) ]Let M, N, P be the intersection point of BC and B C , AB and A B , and CA and C A , respectively., and M, N, P are colinear We need show that AA , BB , CC are concurrent No loss of generality, suppose that AA intersect BB at S We need prove that CC through S Assume that Q is the intersection point of SC and A C , then Q ∈ A C (4) Consider two triangles ABC and A B Q with AA , BB , CQ concurrent at S, we have AB ∩ A B = N BC ∩ B Q = M AC ∩ A Q = P Moreover, AC ∩ A C = P , Q ∈ A C Hence, P ≡ P We see that   N P ∩ BC = M ⇒M ≡M  N P ∩ BC = M Therefore,   BC ∩ B Q = M ⇒Q∈BC  BC ∩ B C = M (5) From (4) and (5), we get: Q ≡ C The the proof is complete 39 Graduation thesis Nguyen Thi Thu Figure 2.6: Pappus’ theorem 2.4 Pappus’ theorem Theorem 2.4.1 Let A, C, E be three collinear points, and B, D, F also collinear points, respectively There exists L, N, M be intersection point of AB and DE, BC and EF , and CD and F A , respectively Then L, N, M are collinear Proof.Let AB ∩ CD = U , CD ∩ EF = V , EF ∩ AB = W Applying Menelaus in the triple point: The triangle U V W has N ∈ V W , C ∈ U V , B ∈ U W and N, B, C 40 Graduation thesis Nguyen Thi Thu are collinear, then we get: −−→ −−→ −→ N V BW CU −−→ −−→ −−→ = N W BU CV (1) The triangle U V W has F ∈ V W , L ∈ U V , A ∈ U W and A, L, F are collinear, then we get: −−→ −−→ −→ F V AW LU −−→ −→ −→ = F W AU LV (2) The triangle U V W has E ∈ V W , C ∈ U V , A ∈ U W and C, A, E are collinear, then we get: −−→ −−→ −→ EV AW CU −−→ −→ −−→ = EW AU CV (3) The triangle U V W has F ∈ V W , D ∈ U V , B ∈ U W and B, D, F are collinear, then we get: −−→ −−→ −−→ F V BW DU −−→ −−→ −−→ = F W BU DV (4) The triangle U V W has E ∈ V W , D ∈ U V , M ∈ U W and B, D, F are collinear, then we get: −−→ −−−→ −−→ EV M W DU −−→ −−→ −−→ = EW M U DV (5) Multiplication side to side of (1), (2), (5) , after then divide of product of (3), (4), we get: −−→ −−−→ −→ N V M W LU −−→ −−→ −→ = N W M U LV 41 Graduation thesis Nguyen Thi Thu Figure 2.7: Exercise 2.7.1 ✷ By Menelaus’ theorem then M, N, L are collinear 2.5 Exercises Exercise 2.5.1 In A2 , Let A, B, C be three independent points, and A1 , B1 , C1 lie on BC, CA, AB, respectively, such that AA1 , BB1 , CC1 are concurrent Prove that BC B1 C1 if and only if A1 is midpoint of BC Solution (Necessary) Assume that BC B1 C1 By Ceva’s theorem we get: [BCA1 ] [CAB1 ] [ABC1 ] = −1 42 (∗) Graduation thesis Nguyen Thi Thu Figure 2.8: Exercise 2.7.2 Since BC B1 C1 , by Thales’ theorem we have [ABC1 ] = [ACB1 ] Hence, [BCA1 ] [CAB1 ] [ACB1 ] = −1, so, [BCA1 ] = −1 Thus A1 is midpoint of B, C (Sufficient) Suppose that A1 is midpoint of BC, i.e, [BCA1 ] = −1 From (∗), we infer [CAB1 ] [ABC1 ] = ⇒ [CAB1 ] = [BAC1 ] Hence, BC B1 C1 Exercise 2.5.2 Let ABC be a triangle, the points X, Y, Z belong to BC, CA, AB, respectively, such that AX, BY, CZ are concurrent at N Asume that AX and Y Z intersect at T , ZX and T B at M, XY and T C at P , and Y Z and BC at Q Prove that M, N, P, Q are collinear Solution We have Y, Z, T are collinear and B, X, C ar collinear we get XZ ∩ T B = M, ZC ∩ Y B = N, T C ∩ XY = P By Pappus’ theorem then M, N, P are collinear 43 Graduation thesis Nguyen Thi Thu Consider two triangle XY Z and T BC, XT, ZC, Y B are concurrent Moreover XZ ∩ T B = M, T C ∩ XY = P, Y Z ∩ BC = Q Hence, by Desargues, M, P, Q are collinear Thus, M, N, P, Q are collinear 44 Graduation thesis Nguyen Thi Thu Conclusions In this thesis, we have presented systematically the following results: (1) Recall the definition and some properties of affine space; affine maps, and affine isomorphism, affine transformation and give some exercises (2) We have presented the single ratio and have proved three theorem in plane : Thales’ theorem, Desargues’ theorem and Pappus’ theorem and give some exercise to apply three theorem This is the first time I got acquainted with scientific research, despite many attempts, I could not avoid may shortcomings Therefore, I look forward to receiving suggestions from teachers and readers I sincerely thanks the teachers and especially Assoc.Prof Nguyen Nang Tam has been dedicated to helping me in the past 45 Bibliography [1 ] Agustí Reventós Tarrida, Affine Maps, Euclidean Motions and Quadrics, pp.1-90 [2 ] Michèle Audin, Geometry, Geometry, Springer,2002, pp.7-32 [3 ] Pham Khac Ban, Pham Binh Do, Hình học afin hình học Ơclit ví dụ tập , NXB Đại học sư phạm ( in Vietnamese) [4 ] Van Nhu Cuong, Ta Man, Hình học afin hình học Ơclit, NXB ĐHQG Hà Nội, 1998 ( in Vietnamese) [5 ] Ha Tram, Bài tập hình học afin hình học Ơclit , NXB ĐHQG Hà Nội, 2005 ( in Vietnamese) [6 ] Nguyen Van Nho, Những định lí chọn lọc hình học phẳng qua kì thi Olympic, NXB Giáo dục , 2001( in Vietnamese) 46 ... topic Affine maps and some theorems in plane geometry for the graduation thesis Aims of the study The purpose of this thesis reseaches into affine mappings and three theorems of affine geometry in. .. 15 1.3.2 Affine transformation 16 1.3.3 Affine mapping in affine coodinates 19 1.3.4 Homological affine mapping 20 1.3.5 Homologicalgline affine mapping ... properties of affine space; affine maps, and affine isomorphism, affine transformation and give some exercises (2) We have presented the single ratio and have proved three theorem in plane : Thales’