Introduction
Prerequisite. Overview.
1 Preliminaries
What is axiomatic approach? What is model? Metric spaces. Examples. Shortcut for distance. Isometries, motions and lines. Half-lines and segments. Angles. Reals modulo 2π. Continuity. Congruent triangles.
Euclidean geometry
2 The Axioms
The Axioms. Lines and half-lines. Zero angle. Straight angle. Vertical angles.
3 Half-planes
Sign of angle. Intermediate value theorem. Same sign lemmas. Half-planes. Triangle with the given sides.
4 Congruent triangles
Side-angle-side condition. Angle-side-angle condition. Isosceles triangles. Side-side-side condition.
5 Perpendicular lines
Right, acute and obtuse angles. Perpendicular bisector. Uniqueness of perpendicular. Reflection. Perpendicular is shortest. Angle bisectors. Circles. Geometric constructions.
6 Parallel lines and similar triangles
Parallel lines. Similar triangles. Pythagorean theorem. Angles of triangle. Transversal property. Parallelograms. Method of coordinates.
7 Triangle geometry
Circumcircle and circumcenter. Altitudes and orthocenter. Medians and centroid. Bisector of triangle. Incenter.
Inversive geometry
8 Inscribed angles
Angle between a tangent line and a chord. Inscribed angle. Inscribed quadrilaterals. Arcs.
9 Inversion
Cross-ratio. Inversive plane and circlines. Ptolemy’s identity. Perpendicular circles. Angles after inversion.
Non-Euclidean geometry
10 Absolute plane
Two angles of triangle. Three angles of triangle. How to prove that something can not be proved? Curvature.
11 Hyperbolic plane
Poincaré disk model. The plan. Auxiliary statements. Axioms: I, II, III, IV, h-V.
12 Geometry of h-plane
Angle of parallelism. Inradius of triangle. Circles, horocycles and equidistants. Hyperbolic triangles. Conformal interpretation.
Incidence geometry
13 Affine geometry
Affine transformations. Constructions with parallel tool and ruler. Matrix form. On inversive transformations.
14 Projective geometry
Real projective plane. Euclidean space. Perspective projection. Projective transformations. Desargues’ theorem.Duality. Axioms.
Additional Topics
15 Spherical geometry
Spheres in the space. Pythagorean theorem. Inversion of the space. Stereographic projection. Central projection.
16 Klein model
Special bijection of h-plane to itself. Klein model. Hyperbolic Pythagorean theorem. Bolyai’s construction.
17 Complex coordinates
Complex numbers. Complex coordinates. Conjugation and absolute value. Euler’s formula. Argument and polar coordinates. Möbius transformations. Elementary transformations. Complex cross-ratio. Schwarz–Pick theorem.
18 Geometric constructions
Classical problems. Constructable numbers. Construction with set square. More impossible constructions.
References
Hints
Index
Used resources