**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 **