1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2019/2020
4. Mathematics - First-cycle studies leading to Bachelor's degree
5. Elementary Geometry

Generate PDF for this page

Elementary Geometry - course description

Aim of the course

Introduction with basic notions, methods of elementary geometry as well as the students' equipment in basic mathematical indispensable tools to formulating and solving typical, the straight lines of tasks and the problems with range of studied direction of studies.

Prerequisites

Save changes

Elementary algebra, linear algebra, basic analytic geometry.

Scope

Lecture
1. Isometrics of Euclidean plane: definitions, examples, kinds, classification of isometrics of plane –2h.
2. The similarity of Euclidean plane: basic definitions, dilatations, classification of similarities of plane - 2h.
3. Affine transformation: basic definitions, property, the analytic figure of affine transformation, matrices criteria - 2h.
4. Points and lines associated with a triangle: Menelaus’ and Ceva’s theorems-2h.
5. The Euler line and the 9-point circle - 2h.
6. The power of point with respect to a circle-2h.
7. The Theorems of Euler. The power straight line of steam of circles. The power centre of three of circles. The Brianchon Theorems -2h.
8. Circle inversion. The Feuerbach's Theorems – 2h.
9. Geometric constructions. Constructional problems, methods of solving the constructional problems. Constructions using ruler and compass - 2h.
10. The impossibility of solving the tree Famous Problems of Antiquity with Euclidean Tools - 2h.
11. Constructions of regular polygons. The constructions of chosen of regular polygons - 2h.
12. Constructions unclassic centres. The Mohr– Marcheroni Construction Theorem. The Poncelet-Steiner Construction Theorem – 2h.
13. Convex Polyhedron, Euler's formula, Platonian clods -2h.
14. An axiomatic approach to Euclidean geometry, absolute geometry. Various formulations of the fifth postulate -2h.
15. Hyperbolic geometry and its models (Klein and Poincaré models). Basic theorems. Other non-Euclidean geometries -2h.

Class
1. Geometric transformations of Euclidean plane: isometrics, method of building the groups of transformations, examples of groups - 2h.
2. Isometrics on plane: analytic formulas central symmetry, translation, axial symmetry, turn around point about directed angle; the group of Isometric of own figures; examples - 2h.
3. Similarity of Euclidean plane: characteristics, examples and classification - 2h.
4. Affine Transformation: analytic figure, matric criteria - 2h.
5. Points and lines associated with a triangle: the Theorems Menelaus and Ceva, Applications of the Theorem of Menelaus and Ceva - 2h.
6. The Euler line and the 9-point circle - geometrical interpretation-2h.
7. Power of a point with respect to a circle – 2.
8. Theorems concerning chords, secants and tangents of a circle. Radical axis of two circles – 2h.
9. Circle inversion. Orthogonal circles- 2h.
10. Interim control written test - 2h.
11. Geometric constructions. Constructional problems, methods of solving the constructional problems. Constructions using the ruler and compass - 2h.
12. The impossibility of solving the tree Famous Problems of Antiquity with Euclidean Tools - 2h.
13. Constructions of regular polygons. The constructions of chosen of regular polygons - 2h.
14. Convex Polyhedron, Eulera's theorem, Platonian clods - 2h.
15. Written test -2h.

Teaching methods

Lecture: conventional, problematic, introduction.
Practice: the classic problematic method, work in groups, the demonstration from explanation, the discussion, storm of brains, work with programme C and R. in computer laboratory.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Assessment form - the final written exam.
The final grades: average grade of tests and exam.
The condition of positive assessment of practices - the positive assessment of two tests as well as the activity on practices.

To pass the test you need to obtain settled (for given test / colloquium) the minimum number of points (50%).
The condition of positive assessment of examination - the positive assessment of multiple choice test (the examples illustrated the lecture) by obtaining the settled minimum number of points of test (50%).
Checking students' preparation to classes as well as their activity on the practices.
The tests (colloquia) include tasks of diverse level of difficulty, permitting to assess if a student has reached the learning outcomes on basic level.

Recommended reading

1. Aleksandrow I. I.: Zbiór geometrycznych zadań konstrukcyjnych, PZWS, Warszawa 1964
2. Borsuk K., Szmielew W.: Podstawy geometrii,. PWN, Warszawa 1970
3. Doman R.: Wykłady z geometrii elementarnej, Wyd. Naukowe UAM, Poznań 2001
4. Kordos,M. Szczerba L., W.: Geometria dla nauczycieli, PWN, Warszawa 1976
5. Coxeter S. M.: Wstep do geometrii dawnej i nowej, PWN, Warszawa 1967
6. Kowalski E.: Geometria dla studentów, WSP, Zielona Góra 1990
7. Modenov P.: Parhomenko A.: Geometric Transformations. Acad. Press, New York, 1965
8. Szmielew W.: Od geometrii afinicznej do euklidesowej, PWN, 1983
9. Zetel S. I.: Geometria trójkąta, PZWS, Warszawa 1964

Further reading

1. Berger M.: Geometrie, Nathan, Paris 1977
2. Coxter H.S.,M, Greitzer S.,L.: Geometry revisited, Toronto New York 1967
3. Neugebauer A.: Wstęp do planimetrii, Wydawnictwo Naukowe US, Szczecin 2000

Notes

Modified by dr Robert Dylewski, prof. UZ (last modification: 20-09-2019 10:25)