Euclidean Plane and its Relatives

This book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalistic.

Geometry Mathematics

Author: Anton Petrunin
Publisher: Pennsylvannia State University
Source: LibreTexts
License: CC BY-SA
Language: EN
Category: Basic Sciences

Read online Download PDF (open licence)

Contents· 150 sections

01: Preliminaries
1.01: What is the axiomatic approach
1.02: What is a model
1.03: Metric spaces
1.04: Shortcut for distance
1.05: Isometries motions and lines
1.06: Half-lines and segments
1.07: Angles
1.08: Reals modulo 2
1.09: Continuity
1.10: Congruent triangles
02: Axioms
2.01: The axioms
2.02: Lines and half-lines
2.03: Zero angle
2.04: Straight angle
2.05: Vertical angles
03: Half-Planes
3.01: Sign of an angle
3.02: Intermediate Value Theorem
3.03: Same sign lemmas
3.04: Half-planes
3.05: Triangle with the given sides
04: Congruent Triangles
4.01: Side-Angle-Side Condition
4.02: Angle-Side-Angle Condition
4.03: Isoseles Triangles
4.04: Side-Side-Side condition
4.05: On side-side-angle and side-angle-angle
05: Perpendicular lines
5.01: Right Acute and Obtuse Angles
5.02: Perpendicular Bisector
5.03: Uniqueness of a perpendicular
5.04: Reflection across a line
5.05: Perpendicular is shortest
5.06: Circles
5.07: Geometric constructions
06: Similar triangles
6.01: Similar triangles
6.02: Pythagorean theorem
6.03: Method of similar triangles
6.04: Ptolemy's inequality
07: Parallel Lines
7.01: Parallel lines
7.02: Reflection Across a Point
7.03: Transversal Property
7.04: Angles of triangles
7.05: Parallelograms
7.06: Method of coordinates
7.07: Apollonian Circle
08: Triangle Geometry
8.01: Circumcircle and circumcenter
8.02: Altitudes and orthocenter
8.03: Medians and centroid
8.04: Angle Bisectors
8.05: Equidistant Property
8.06: Incenter
8.07: More exercises
09: Inscribed angles
9.01: Angle between a tangent line and a chord
9.02: Inscribed angle
9.03: Points on a common circle
9.04: Method of additional circle
9.05: Arcs of circlines
9.06: Tangent half-lines
10: Inversion
10.01: Inversion
10.02: Cross-ratio
10.03: Inversive plane and circlines
10.04: Method of inversion
10.05: Perpendicular circles
10.06: Angles after inversion
11: Neutral plane
11.01: Neutral plane
11.02: Two angles of a triangle
11.03: Three angles of triangle
11.04: Defect
11.05: How to prove that something cannot be proved
11.06: Curvature
12: Hyperbolic Lane
12.01: Conformal Disc Model
12.02: Plan of the proof
12.03: Auxiliary statements
12.04: Axiom I
12.05: Axiom II
12.06: Axiom III
12.07: Axiom IV
12.08: Axiom h-V
12.09: Hyperbolic trigonometry
13: Geometry of the h-plane
13.01: Angle of parallelism
13.02: Inradius of h-triangle
13.03: Circles horocycles and equidistants
13.04: Hyperbolic triangles
13.05: Conformal interpretation
13.06: Pythagorean theorem
14: Affine Geometry
14.01: Affine transformations
14.02: Constructions
14.03: Fundamental Theorem of Affine Geometry
14.04: Algebraic lemma
14.05: On inversive transformations
15: Projective Geometry
15.01: Projective Completion
15.02: Euclidean Space
15.03: Space Model
15.04: Perspective projection
15.05: Projective transformations
15.06: Moving points to infinity
15.07: Duality
15.08: Construction of a polar
15.09: Axioms
16: Spherical geometry
16.01: Euclidean space
16.02: Pythagorean Theorem
16.03: Section 3-
16.04: Section 4-
16.05: Section 5-
17: Projective Model
17.01: Section 1-
17.02: Section 2-
17.03: Section 3-
18: Complex Coordinates
18.01: Section 1-
18.02: Section 2-
18.03: Section 3-
18.04: Section 4-
18.05: Section 5-
18.06: Section 6-
18.07: 18.7-
18.08: 18.8-
18.9: Complex cross-ratio
18.10: Schwarz-Pick theorem
19: Geometric Constructions
19.01: Section 1-
19.02: Section 2-
19.03: Section 3-
19.04: Section 4-
19.05: Section 5-
20: Area
20.01: Section 1-
20.02: Section 2-
20.03: Section 3-
20.04: Section 4-
20.05: Section 5-
20.06: Section 6-
20.07: 20.7-
20.08: 20.8-
20.9: Area in the neutral planes and spheres
20.10: Quadrable sets