1. komponenta

Lecture type	Total
Lectures	30
Exercises	30

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: It is an introductory geometrical course in this study and some kind of a "bridge" course from the secondary school and the so-called university mathematics. The main aim is to systematize and enhance the understanding of elementary (synthetic) geometry. A lot of topics will be presented by means of dynamic geometry software.


COURSE DESCRIPTION AND SYLLABUS:
Plane geometry
Introduction. Fundamental objects in a plane (points and straight lines). Axioms of an Euclidean plane geometry (on elementary level). Euclid's parallel postulate.
Point sets in plane. Half-line. Segment. Convex set. Half-plane. Angle. Measure of angle. Vertically opposite angles. Angles with parallel sides. Angles with perpendicular sides. Angles on transversal. Triangle. Angle sum theorem. Quadrilateral and its diagonals. Trapezoid. Parallelogram. Rhombus. Rectangular. Square. Quadrilateral with perpendicular diagonals. Polygons. Circle and circumference.
Congruence of triangles. Definition and theorems on congruence of triangles. Four characteristic points in triangle. Properties of parallelogram and rhomboid. Theorem on midline of a triangle. Inscribed and circumscribed circle of a triangle. Theorem on median of a trapezoid. Theorem on bisectors of an angle.
Perimeter and area. Perimeter and area of polygons and above-mentioned quadrilaterals Heron formula. Escribed circle of triangle. Connection between area of a triangle, sides and radii of escribed circles. Area of a circle. Perimeter of a circumference.
Similarity of triangles. The Thales triangle figure. Theorem on angle bisector in a triangle. Definition and theorems of similarity of triangles. Pythagorean theorem and its converse. Euclid's theorem. Ceva's and Menelaus' theorems.
Theorems involving circle. Theorem on peripherical and central angle. Circumsribed quadrilatelar. Quadrilateral inscribable in a circle. Power of a point with respect to the circle. Euler's theorem. Feuerbach's theorem.
Application of trigonometry in triangle. Trigonometrical functions of angle. Trigonometry in a right-angled triangle. Law of sines. Cosine law. Brahmagupta's theorem.
Plane transformations. Isometries. Reflection with respect to a line and a point. Rotation. Translation. Euler's line. Similarity and homothety. Inversion. Ptolemy's theorem.
Second-degree curves. Ellipse. Hyperbola. Circles connected with ellipse and hyperbola. Parabola (non-analytic approach).

Stereometry.
Introduction. Fundamental objects of space (points, straight lines and planes). Axioms of Euclidean geometry. Half-space. Relation "to be parallel" and "to be perpendicular" between straight lines and planes. Theorem on three perpendiculars.
Angles. Angle between two straight lines. Angle between a line and a plane. Angle between two planes.
Distances. Distance from a point to a plane and a line. Distance between two lines in a space. Symmetry plane of a segment. Symmetry plane of two planes. Dihedrons.
Space transformations. Translation of space. Reflection of space with respect to the point. Reflection of space with respect to a plane and a line. Rotation. Homothety and similarity transformations of space.
Polyhedra. Types of polyhedra. Euler's theorem for polyhedra. Regular polyhedra.
Cone. Cylinder. Sphere. Ball.
Surface area and volume of solids. Area and volume of prism, pyramid and truncated pyramid. Theorem about hedgehog for polyhedrons. Principle of Cavalieri. Surface area and volume of cylinder, cone and ball. Guldin's rules for volumen and surface area.