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Topics of Geometry

Topics of Geometry

Code	10828
Year	1
Semester	S1
ECTS Credits	12
Workload	TP(45H)
Scientific area	Mathematics
Entry requirements	-
Mode of delivery	Face-to-face.
Learning outcomes	To provide a historical and integrated perspective of the development of the main areas of elementar geometry. 1. To provide a historical perspective of the development of geometry. 2. To deepen the understanding of some topics of Euclidean geometry. 3. To understand the fundamentals and major applications of inversive and Projective Geometry. 4. To understand the main results on equidecomposability of polygons and polyhedra. 5. To establish links between the contents of this course and the curricula of Mathematics in the 3rd cycle of Basic Education and Secondary Education.on and Secondary Education.
Syllabus	I- A Historical Introduction: the geometry in Mesopotamia and Egypt; Euclid's Elements; Descartes; Gauss; axiom of parallels and non-Euclidean geometries; Erlangen program. II- Euclidean geometry: geometry of the triangle; theorems of Ceva and Menelaus; notable points and lines; geometry of the circle; power of a point; radical axis; Simson lines; Ptolemy's Theorem; Morley's Theorem the use of dilations and isometries in solving geometric problems. III- Inversive Geometry: inversions and inversive plane; cross-ratio: Pappus Arbelos; problem of Appolonius; Steiner’s porism; Peaucellier mechanism. IV- Projective geometry: projective plane; principle of duality; projective transformations; Desargues Theorem; Fundamental Theorem of projective geometry; Pappus theorem; polarities and conics; Theorems of Pascal and Brianchon. V- Polygons and polyhedra: Euler's formula; Pick's Theorem; Platonic solids); equidecomposable polygons; Theorem Wallace-Bolyai-Gerwian; Hilbert’s third problem.
Main Bibliography	1) Aigner, M., Ziegler, G. M., Proofs from the Book, Springer, 2001. 2) Araújo, P. Curso de Geometria, Gradiva, 1999. 3) Coxeter, H., Greitzer, S., Geometry revisited, New Mathematical Library, AMS,1967. 4) Elon Lages Lima, Matemática e Ensino, Gradiva, 2004. 5) Ogilvy, S., Excursions in Geometry, Dover, 1969. 6) Yaglom, I.M., Geometric Transformations I,II,III, New Mathematical Library, AMS, 1968.
Language	Portuguese. Tutorial support is available in English.