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Specht, Et Al: Euclidean Geometry Exercises and Answers: Contents Specht, et al: Euclidean Geometry Exercises and Answers: Contents Introduction to Exercises and Answers .................... 1 Chapter 1: Preliminaries and Incidence Geometry (I)........ 1 Chapter 2: Incidence with Parallelism (IP) . 5 Chapter 3: Collineations of an Affine Plane (CAP).......... 8 Chapter 4: Incidence and Betweenness (IB) ................ 10 Chapter 5: Pasch Geometry (PSH) ....................... 14 Chapter 6: Ordering a Line in a Pasch Plane (ORD) ........ 34 Chapter 7—has no Exercises (COBE)..................... 38 Chapter 8: Neutral Geometry (NEUT) .................... 38 Chapter 9: Free Segments of a Neutral Plane (FSEG) ....... 79 Chapter 10: Rotations about a Point of a Neutral Plane (ROT) 83 Chapter 11: Euclidean Geometry Basics (EUC) ............ 90 Chapter 12: Isometries of a Euclidean Plane (ISM) ......... 93 Chapter 13: Dilations of a Euclidean Plane (DLN) . 96 Chapter 14: A Line as an Ordered Field (OF).............. 100 Chapter 15: Similarity on a Euclidean Plane (SIM) . 103 Chapter 16: Axial Affinities of a Euclidean Plane (AX) . 106 Chapter 17: Rational Points on a Line (QX) ............... 108 Chapter 18: A Line as Real Numbers (REAL); Coordinatization of a Plane (RR)......................... 110 Chapter 19—has no Exercises (AA)....................... 115 Chapter 20: Ratios of Sensed Segments (RS) ............... 115 Chapter 21: Consistency and Independence of Axioms (LM)(MLT) .......................................... 116 Chapter 1: Preliminaries and Incidence Geometry (I) 1 Introduction to Exercises and Answers This file contains answers (solutions) to exercises in Euclidean Geometry and its Subgeometries by Specht, Jones, Calkins, and Rhoads (Birkh¨auser, 2015). All references to chapters, theorems, definitions, remarks, and figures refer to this work. Most exercises are in the form of statements, so that “answers” are actually proofs, and are labeled as such. Here we state all the exercises for each chapter, but give solutions only to those that are starred; most of these are needed for the development of the book. Thus, a solution is given for Exercise I.13* but not for Exercise I.14. We encourage you to read each exercise, make a sketch if it helps you to visualize it, and get in mind how to prove it, even if you don’t actually put the details together. You can justify skipping an occasional exercise only if you are quite sure you could construct the proof if you had to, and feel it is a waste of your time to supply all the details. But beware that supplying all the details may look deceptively simple when you give a theorem a cursory glance. It is possible that you may create new solutions for exercises that are more elegant than the ones we have given. The authors will appreciate receiving any such improvements, as well as corrections to errors you may find in the proofs given. Chapter 1: Exercises and Answers for Preliminaries and Incidence Geometry (I) Exercise I.1* If L and M are distinct lines and if L ∩ M 6= ∅, then L∩M is a singleton. Exercise I.1 Proof. Assume L∩M has two distinct members A and B; then each of the points A or B belongs to L and to M. By Axiom I.1 L = M, contradicting the given fact that L and M are distinct. Hence our assump- tion that A 6= B is false, and since L ∩ M 6= ∅, L∩M is a singleton. ⊓⊔ Exercise I.2* (A) If A and B are distinct points, and if C and D are ←→ ←→ ←→ distinct points on AB, then CD = AB. 2 (B) If A, B, and C are noncollinear points, and if D, E, and F are non- ←−−→ ←−−→ ←−−→ collinear points on ABC, then DEF = ABC. Exercise I.2 Proof. (A) Since C and D are distinct points, by Axiom I.1, ←→ ←→ CD = AB. ←−−→ (B) Since D, E, and F are noncollinear points on ABC, by Axiom I.2, ←−−→ ←−−→ DEF = ABC. ⊓⊔ Exercise I.3* If L and M are lines and L⊆M, then L = M. Exercise I.3 Proof. By Axiom I.5(A) there exist distinct points A and B on L. Since L⊆M, A and B are on M. By Axiom I.1 L = M. ⊓⊔ Exercise I.4* Let A and B be two distinct points, and let D, E, and ←→ F be three noncollinear points. If AB contains only one of the points D, E, ←→ ←→ ←→ ←→ and F , then each of the lines DE, EF , and DF intersects AB in at most one point. ←→ Exercise I.4 Proof. We may choose our notation so that AB contains ←→ ←→ ←→ the point D. Then if DE (or DF ) intersects AB in two points, by Exercise ←→ ←→ ←→ ←→ I.2(A) DE = AB, (or DF = AB) and both D and E (D and F ) are members ←→ of AB, contradicting the assumption that this line contains only one of the ←→ ←→ ←→ points D, E, or F . Since E and F do not belong to AB, EF 6= AB and by Exercise I.1, if these intersect they intersect in a singleton. ⊓⊔ Exercise I.5* If E is a plane, L is a line such that E ∩ L 6= ∅, and L is not contained in E, then E ∩ L is a singleton. Exercise I.5 Proof. Since E ∩ L 6= ∅, there exists a point A belonging to E and to L. Assume there exists a point B distinct from A which belongs to ←→ E and to L. By Axiom I.1 L = AB and by Axiom I.3 L⊆E. This contra- dicts the given fact that L is not contained in E. Hence E ∩L is a singleton. ⊓⊔ Exercise I.6 Let D and E be distinct planes such that D ∩E 6= ∅, so that (by Theorem I.4) D∩E is a line L; let P be a point on D but not on L; and ←→ let Q be a point on E but not on L. Then PQ and L are not coplanar. Exercise I.7* Given a line L and a point A not on L, there exists one and only one plane E such that A ∈E and L⊆E. Chapter 1: Preliminaries and Incidence Geometry (I) 3 Exercise I.7 Proof. By Axiom I.5 there are two distinct points on L, and by assumption these are not collinear with A. By Axiom I.2 there exists ex- actly one plane E containing all three points. By Axiom I.3 L⊆E. ⊓⊔ Exercise I.8* Let A, B, C, and D be noncoplanar points. Then each of the triples {A, B, C}, {A,B,D}, {A,C,D}, and {B,C,D} is noncollinear. Exercise I.8 Proof. If one of the triples is collinear, all of its points belong to a line L. By Exercise I.7 there exists a unique plane E such that L and the one point among A, B, C, and D which is not in the triple are in that plane, hence the four points are coplanar. ⊓⊔ Exercise I.9 There exist four distinct planes such that no point is com- mon to all of them. Exercise I.10 Every plane E contains at least three lines L, M, and N such that L∩M∩N = ∅. Exercise I.11 Every plane contains (at least) three distinct lines. Exercise I.12 Space contains (at least) six distinct lines. Exercise I.13* If L is a line contained in a plane E, then there exists a point A belonging to E but not belonging to L. Exercise I.13 Proof. Axiom I.5 says that there exist at least three non- collinear points on E. Therefore not all of these points can be in any line L; let A be that point from among these three that is not on this line. ⊓⊔ Exercise I.14 If P is a point in a plane E, then there is a line L such that P ∈ L and L⊆E. Exercise I.15 If a plane E has (exactly) three points, then each line con- tained in E has (exactly) two points. Exercise I.16 If a plane E has exactly four points, and if all of the lines contained in E have the same number of points, then each line contained in E has (exactly) two points. 4 Exercise I.17 If each line in space has at least three points, then: (1) Each point of a plane is a member of at least three lines of the plane; (2) Each plane has at least seven points; (3) Each plane contains at least seven lines. Exercise I.18 In this exercise we will use the symbolism “PkQ” to indi- cate that two planes P and Q do not intersect. Consider what can happen if the restrictions of P1 ∩P2, P1 ∩P3, and P2 ∩P3 being nonempty are removed in Theorem I.9. Sketch at least four possibilities (P1 kP2 kP3, P1 = P2 kP3, P1 = P2 = P3, P1 ∩P2 = ∅, but P1 ∩P3 = L2 and P2 ∩P3 = L1) and deter- mine if these can be proved within incidence geometry. Exercise I.19 Count the number of lines in the 8-point model. Compare n(n+1) n this with T = 2 , triangular numbers, for n = 7. Compare it also with n! n r C = r!(n−r)! , the number of combinations of n items taken r at a time, where n = 8 and r = 2. Exercise I.20 Count the number of planes in the 8-point model. Com- pare this with nCr for n = 8 and r = 3. Note the reduction by a factor of four due to the fact that each plane has four points. Can you form a similar argument with r = 4? Exercise I.21 Consider a 4-point model with the four points configured like the vertices of a tetrahedron.
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