Euclidean Plane and Its Relatives꞉ A Minimalist Introduction

Description

"This book is meant to be rigorous, conservative, elementary, and mini- malist. At the same time, it includes about the maximum that 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." Contents Introduction A. Prerequisite B. Overview C. Disclaimer D. Recommended resources E. Acknowledgments 1. Preliminaries A. What is the axiomatic approach? B. What is a model? C. Metric spaces D. Shortcut for distance E. Isometries, motions, and lines F. Half-lines and segments G. Angles H. Reals modulo 2·π I. Continuity J. Congruent triangles Euclidean Geometry 2. Axioms A. The axioms B. Lines and half-lines C. Zero angle D. Straight angle E. Vertical angles 3. Half-planes A. Sign of an angle B. Intermediate value theorem C. Same sign lemmas D. Half-planes E. Triangle with the given sides 4. Congruent triangles A. Side-angle-side B. Angle-side-angle C. Isosceles triangles D. Side-side-side E. On side-side-angle and side-angle-angle 5. Perpendicular lines A. Right, acute and obtuse angles B. Perpendicular bisector C. Uniqueness of a perpendicular D. Reflection across a line E. Direct and indirect motions F. Perpendicular is shortest G. Circles H. Geometric constructions 6. Similar triangles A. Similar triangles B. Pythagorean theorem C. Method of similar triangles D. Ptolemy's inequality 7. Parallel lines A. Parallel lines B. Reflection across a point C. Transversal property D. Angles of triangles E. Parallelograms F. Method of coordinates G. Apollonian circle 8. Triangle geometry A. Circumcircle and circumcenter B. Altitudes and orthocenter C. Medians and centroid D. Angle bisectors E. Equidistant property F. Incenter Inversive geometry 9. Inscribed angles A. Angle between a tangent line and a chord B. Inscribed angle C. Points on a circle D. Method of additional circle E. Arcs of circlines F. Tangent half-lines 10. Inversion A. Cross-ratio B. Inversive plane and circlines C. Method of inversion D. Perpendicular circles E. Angles after inversion Non-Euclidean Geometry 11. Neutral plane A. Two angles of a triangle B. Three angles of triangle C. Defect D. Proving that something cannot be proved E. Curvature 12. Hyperbolic plane A. Conformal disc model B. Plan of the proof C. Auxiliary statements D. Axioms E. Hyperbolic trigonometry 13. Geometry of the h-plane A. Angle of parallelism B. Inradius of h-triangle C. Circles, horocycles, and equidistants D. Hyperbolic triangles E. Conformal interpretation F. Pythagorean theorem Additional topics 14. Affine geometry A. Affine transformations B. Constructions C. Fundamental theorem of affine geometry D. Algebraic lemma E. On inversive transformations 15. Projective geometry A. Projective completion B. Euclidean space C. Model of space D. Perspective projection E. Projective transformations F. Moving points to infinity G. Duality H. Construction of a polar I. Axioms 16. Spherical geometry A. Euclidean space B. Pythagorean theorem C. Inversion D. Stereographic projection E. Central projection 17. Projective model A. Special bijection on the h-plane B. Projective model C. Bolyai's construction 18. Complex coordinates A. Complex numbers B. Complex coordinates C. Conjugation and absolute value D. Euler's formula E. Argument and polar coordinates F. Method of complex coordinates G. Fractional linear transformations H. Elementary transformations I. Complex cross-ratio J. Schwarz–Pick theorem 19. Geometric constructions A. Classical problems B. Impossible constructions C. Constructible numbers D. Set-square constructions E. Verifications F. Comparison of construction tools 20. Area A. Solid triangles B. Polygonal sets C. Definition of area D. Vanishing area and subdivisions E. Rectangles F. Parallelograms G. Triangles H. Area method I. Neutral planes and spheres J. Quadrable sets References Hints Index Used resources