Objectives and competences
The objective of the course is to present key analytical environments suitable for use in geometry (Cartesian coordinates, homogeneous Cartesian coordinates, complex numbers, trilinear coordinates). In each of these environments problems will be presented in solving which this environment is very effective. The objective of the course is also to show (sometimes unexpected) interweaving of different branches of mathematics.
Content (Syllabus outline)
• Analytic geometry in Cartesian coordinates. Lines, conics. Examples of use in geometry. Euler's conics.
• Analytical geometry in homogeneous Cartesian coordinates. Projective plane. Conics. Joachimstahl's equation, tangent, pole, polar.
• Analytical geometry in trilinear coordinates. Examples of use. Euler line, Kiepert hyperbola. Isogonal transformation. Cubics associated with a triangle. Kimberling’s definition of a triangle center.
• Complex numbers in geometry. Necessary and sufficient conditions for similarity of triangles with given vertices. Conditions that three given points are the vertices of an equilateral triangle. Napoleon and Thebaultov theorem. Napoleon – Barlotti theorem. Colinearity and concyclity. Ptolemey theorem. Clifford theorems.
Learning and teaching methods
• Lectures
• Theoretical exercises
• Individual work
• Teaching and learning are done through the didactic use of ICT
Intended learning outcomes - knowledge and understanding
On completion of this course a student will be able to
• Master the key methods of work in the most common analytical environments suitable for use in geometry (Cartesian coordinates, homogeneous Cartesian coordinates, complex numbers, trilinear coordinates).
• Given the nature of the geometric problem, he will be able to choose the most favorable of these environments and use his means.
• Understand the advantages of dealing with geometric problems in a projective plane and mastering work in it through various types of homogeneous coordinates.
• Comprehend the interweaving of different branches of mathematics (linear algebra, symmetric polynomials, complex numbers).
• Describe some recent results in triangle geometry and to illustrate them with computer programs for dynamic geometry.
Intended learning outcomes - transferable/key skills and other attributes
• Creativity: Awareness of the innovativeness of mathematicians in overcoming the limitations of certain mathematical environments by creating new environments (a projective plane, trilinear coordinates).
• Problem solving: Awareness that specific tools can be extremely effective in solving certain problems, but can be completely inappropriate for solving other types of problems.
• Technical skills: Using computer programs (for dynamic geometry, for expressing symmetric polynomials with elementary symmetric polynomials etc.).
Readings
? B. Spain: Analytical conics, Dover Publications, Mineola, New York, 2007.
? O. Botema, R. Erne, R. Hartshorne: Topics in elementary geometry, Springer, New York, 2008
Liang-shin Hahn: Complex numbers & geometry, MAA, Washington, 1994
Additional information on implementation and assessment Exams:
Written exam – problems
Oral exam
Each of the mentioned assessments must be assessed with a passing grade.
Passing grade of written exam – problems is required to take the oral exam.
