Second Level Degree in Mathematics
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COMPLEMENTARY MATHEMATICS - 6 CFU
Teacher
Francesco Ciraulo
Scheduled Period
I Year - 2 Semester | 28/02/2022 - 11/06/2022
Hours: 48 (24 esercitazione, 24 lezione)
Prerequisites
Basics of Euclidean geometry, linear algebra, group theory.
Target skills and knowledge
The general aim of the course is to provide an "advanced" knowledge of Euclidean geometry, with particular emphasis on a synthetic approach and with attention to historical, epistemological and didactic aspects.
In particular, the knowledge, skills and competences expected by the students are the following:
- knowing the main results of the modern(from the 18th century onwards) geometry of the triangle;
- being aware of the historical development of the discipline, also in view of a possible didactic transposition;
- mastering the synthetic approach to geometry;
- knowing how to use geometric transformations, in particular homotheties;
- knowing how to connect the objectives of the Italian curriculum for secondary school with the contents of the course;
- knowing how to use the dynamic geometry software Geogebra, including its main 3D features, also in view of a possible didactic use.
Examination methods
Oral exam.
The exam will begin with a presentation of a project developed in a dynamic geometry environment.
The exam is aimed at verifying the achievement of the expected objectives, both in terms of disciplinary content and of maturity and awareness, also in view of a possible didactic transposition.
Assessment criteria
The following items will be evaluated:
- the formal correctness of the proof of the theorems,
- the ability to apply the knowledge acquired to solving exercises and problems,
- the completeness of the acquired knowledge,
- the rigor in the use of geometric terminology,
- the ability to use dynamic geometry software.
Course contents
Plane transformations, isometries, similarities, circle inversion. Basics of sapce isometries (via geometric software).
Triangles and their main centers. Euler's line. Medial triangle and orthic triangle.
Nine-point circle. Tritangent circles. Feuerbach's theorem.
Power of a point with respect to a circle. Euler's theorem.
The angle bisector theorem. The circle of Apollonius.
Ceva's and Menelaus's theorems. Pappus's, Pascal's, and Brianchon's theorems.
Fermat point, Napoleon triangle. Morley triangle.
Geometry of paper folding.
Conics as envelopes.
Cyclic quadrangles. Ptolemy's theorem.
Platonic solids and their symmetries (via geometric software).
Planned learning activities and teaching methods
Lectures and exercises with the participation of students.
Some lessons will involve the use of dynamic geometry software.
Additional notes about suggested reading
In addition to the reference textbook, the following additional book are recommended.
- Coxeter and Greitzer, Geometry revisited, The Mathematical Association of America, 1967.
- Dedò, Geometric transformations, Decibel-Zanichelli, 1996.
- Posamentier, Advanced Euclidean Geometry, John Wiley & Sons, 2002.