Euclidean plane and its relatives A minimalist introduction Anton Petrunin arXiv:1302.1630v19 [math.HO] 21 May 2021 This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/4.0/ Contents Introduction 6 Prerequisite. Overview. 1 Preliminaries 9 What is the axiomatic approach? What is a model? Metric spaces. Examples. Shortcut for distance. Isometries, motions, and lines. Half-lines and segments. Angles. Reals modulo 2 π. Continuity. Congruent triangles. · Euclidean geometry 2 Axioms 18 The axioms. Lines and half-lines. Zero angle. Straight angle. Vertical angles. 3 Half-planes 23 Sign of an angle. Intermediate value theorem. Same sign lemmas. Half-planes. Triangle with the given sides. 4 Congruent triangles 30 Side-angle-side condition. Angle-side-angle condition. Isosce- les triangles. Side-side-side condition. On side-side-angle and side-angle-angle. 5 Perpendicular lines 34 Right, acute and obtuse angles. Perpendicular bisector. Uniqueness of perpendicular lines. Reflection across a line. Perpendicular is shortest. Circles. Geometric constructions. 6 Similar triangles 42 Similar triangles. Pythagorean theorem. Method of similar triangles. Ptolemy’s inequality. 3 4 CONTENTS 7 Parallel lines 47 Parallel lines. Reflection across a point. Transversal property. Angles of triangle. Parallelograms. Method of coordinates. Apollonian circle. 8 Triangle geometry 56 Circumcircle and circumcenter. Altitudes and orthocenter. Medians and centroid. Angle bisectors. Equidistant property. Incenter. Inversive geometry 9 Inscribed angles 63 Angle between a tangent line and a chord. Inscribed angle. Points on a common circle. Method of additional circle. Arcs of circlines. Tangent half-lines. 10 Inversion 71 Cross-ratio. Inversive plane and circlines. Method of inver- sion. Perpendicular circles. Angles after inversion. Non-Euclidean geometry 11 Neutral plane 81 Two angles of triangle. Three angles of triangle. Defect. How to prove that something cannot be proved. Curvature. 12 Hyperbolic plane 89 Conformal disc model. Plan of the proof. Auxiliary state- ments. Axioms: I, II, III, IV, h-V. Hyperbolic trigonometry 13 Geometry of the h-plane 101 Angle of parallelism. Inradius of triangle. Circles, horocycles, and equidistants. Hyperbolic triangles. Conformal interpre- tation. Pythagorean theorem. Additional topics 14 Affine geometry 112 Affine transformations. Constructions. Fundamental theorem of affine geometry. Algebraic lemma. On inversive transfor- mations. CONTENTS 5 15 Projective geometry 119 Projective completion. Euclidean space. Space model. Per- spective projection. Projective transformations. Moving points to infinity. Duality. Axioms. 16 Spherical geometry 130 Euclidean space. Pythagorean theorem. Inversion of the space. Stereographic projection. Central projection. 17 Projective model 136 Special bijection on the h-plane. Projective model. Bolyai’s construction. 18 Complex coordinates 143 Complex numbers. Complex coordinates. Conjugation and absolute value. Euler’s formula. Argument and polar co- ordinates. Method of complex coordinates. Fractional lin- ear transformations. Elementary transformations. Complex cross-ratio. Schwarz–Pick theorem. 19 Geometric constructions 154 Classical problems. Constructible numbers. Constructions with a set-square. Verifications. Comparison of construction tools. 20 Area 162 Solid triangles. Polygonal sets. Definition of area. Vanishing area and subdivisions. Area of solid rectangles, parallelograms and triangles. Area method. Area in the neutral planes and spheres. Quadrable sets. References Hints 176 Index 201 Used resources 204 Introduction This book is meant to be rigorous, conservative, elementary, and mini- malist. At the same time, it includes about the maximum what students can absorb in one semester. Approximately one-third of the material used to be covered in high school, but not anymore. The present book is based on the courses given by the author at the Pennsylvania State University as an introduction to the foundations of geometry. The lectures were oriented to sophomore and senior university students. These students already had a calculus course. In particular, they are familiar with real numbers and continuity. It makes it possible to cover the material faster and in a more rigorous way than it could be done in high school. Prerequisite The students should be familiar with the following topics: Elementary set theory: , , , , , . ⋄ ∈ ∪ ∩ \ ⊂ × Real numbers: intervals, inequalities, algebraic identities. ⋄ Limits, continuous functions, and the intermediate value theorem. ⋄ Standard functions: absolute value, natural logarithm, exponential ⋄ function. Occasionally, trigonometric functions are used, but these parts can be ignored. Chapter 14 uses basic vector algebra. ⋄ To read Chapter 16, it is better to have some previous experience ⋄ with the scalar product, also known as dot product. To read Chapter 18, it is better to have some previous experience ⋄ with complex numbers. 6 CONTENTS 7 Overview We use the so-called metric approach introduced by Birkhoff. It means that we define the Euclidean plane as a metric space that satisfies a list of properties (axioms). This way we minimize the tedious parts which are unavoidable in the more classical Hilbert’s approach. At the same time, the students have a chance to learn basic geometry of metric spaces. Here is a dependency graph of the chapters. 1 2 3 4 13 12 11 5 6 17 16 10 9 7 8 15 14 18 19 20 In (1) we give all the definitions necessary to formulate the axioms; it includes metric space, lines, angle measure, continuous maps, and con- gruent triangles. Further, we do Euclidean geometry: (2) Axioms and immediate corol- laries; (3) Half-planes and continuity; (4) Congruent triangles; (5) Circles, motions, perpendicular lines; (6) Similar triangles and (7) Parallel lines these are the first two chapters where we use Axiom V, an equivalent of Euclid’s parallel postulate. In (8) we give the most classical theorem of triangle geometry; this chapter is included mainly as an illustration. In the following two chapters we discuss the geometry of circles on the Euclidean plane: (9) Inscribed angles; (10) Inversion. It will be used to construct the model of the hyperbolic plane. Further, we discuss non-Euclidean geometry: (11) Neutral geometry geometry without the parallel postulate; (12) Conformal disc model this is a construction of the hyperbolic plane, an example of a neutral plane which is not Euclidean. In (13) we discuss geometry of the constructed hyperbolic plane this is the highest point in the book. In the remaining chapters, we discuss some additional topics: (14) Affine geometry; (15) Projective geometry; (16) Spherical geometry; (17) Projective model of the hyperbolic plane; (18) Complex coordinates; (19) Geometric constructions; (20) Area. The proofs in these chapters are not completely rigorous. We encourage the use of visual assignments at the author’s website. 8 CONTENTS Disclaimer It is impossible to find the original reference to most of the theorems discussed here, so I do not even try to. Most of the proofs discussed in the book already appeared in Euclid’s Elements. Recommended books Byrne’s Euclid [7] a colored version of the first six books of Eu- ⋄ clid’s Elements edited by Oliver Byrne. Kiselev’s textbook [10] a classical book for school students; it ⋄ should help if you have trouble following this book. Hadamard’s book [9] an encyclopedia of elementary geometry ⋄ originally written for school teachers. Prasolov’s book [15] is perfect to master your problem-solving skills. ⋄ Akopyan’s book [1] a collection of problems formulated in figures. ⋄ Methodologically my lectures were very close to Sharygin’s textbook ⋄ [17]. This is the greatest textbook in geometry for school students, I recommend it to anyone who can read Russian. Acknowledgments Let me thank Thomas Barthelme, Matthew Chao, Svetlana Katok, Alexander Lytchak, Alexei Novikov, and Lukeria Petrunina for useful suggestions and correcting misprints. This work was partially supported by NSF grants DMS-0103957, DMS-0406482, DMS-0905138, DMS-1309340, DMS-2005279, and Simons Foundation grants 245094, 584781. Chapter 1 Preliminaries What is the axiomatic approach? In the axiomatic approach, one defines the plane as anything that satis- fies a given list of properties. These properties are called axioms. The axiomatic system for the theory is like the rules for a game. Once the axiom system is fixed, a statement is considered to be true if it follows from the axioms and nothing else is considered to be true. The formulations of the first axioms were not rigorous at all. For example, Euclid described a line as breadthless length and a straight line as a line that lies evenly with the points on itself. On the other hand, these formulations were sufficiently clear so that one mathematician could understand the other. The best way to understand an axiomatic system is to make one by yourself. Look around and choose a physical model of the Euclidean plane; imagine an infinite and perfect surface of a chalkboard. Now try to collect the key observations about this model. Assume for now that we have an intuitive understanding of such notions as line and point. (i) We can measure distances between points. (ii) We can draw a unique line that passes thru two given points. (iii) We can measure angles. (iv) If we rotate or shift we will not see the difference. (v) If we change the