Geometry

Code:

M152

Acronym:

M152

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2013/2014 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AST	1	Plano de Estudos a partir de 2008	1	-	7,5	-	202,5
			3
L:B	0	Plano de estudos a partir de 2008	3	-	7,5	-	202,5
L:CC	0	Plano de estudos de 2008 até 2013/14	3	-	7,5	-	202,5
L:F	0	Plano de estudos a partir de 2008	3	-	7,5	-	202,5
L:G	0	P.E - estudantes com 1ª matricula anterior a 09/10	3	-	7,5	-	202,5
		P.E - estudantes com 1ª matricula em 09/10	3	-	7,5	-	202,5
L:M	31	Plano de estudos a partir de 2009	1	-	7,5	-	202,5
			2
			3
L:Q	0	Plano de estudos Oficial	3	-	7,5	-	202,5

Teaching language

Suitable for English-speaking students

Objectives

With this course, it is intended that students will know and understand some of the main results of Euclidean geometry that, for its historical importance, should be of general knowledge for mathematicians. In this course the students should also develop their ability to solve geometric problems and to be able to visualize in space.

Learning outcomes and competences

Increased ability to understand and use classical results from geometry. Increased ability to solve problems in a "non-systematic" way. Better visualization of space. Better understanding of space.

Working method

Presencial

Program

Brief reference to a set of axioms (of the type suggested by Birkhoff). Reference to ruler and compass constructions and study of some initial constructions. Notion of congruence and the cases of congruence of triangles. Isometries of the plane. The theorem of Thales. Notion of Area. The Pythagorean theorem. Theorems of Ceva, Menelaus and the angle bisector theorem. Triangle centers. Cases of similarity of triangles and similarity transformations. Construction of the fourth proportional and the mean proportional. Law of sines and law of cosines. Homotheties. Euler's line. Study of the circumference: the inscribed angle theorem; power of a point relative to a circumference. Radical axis of two circles. Introduction and study of the concept of inversion. Applications to Apolonius' Problem.

Mandatory literature

Araújo Paulo Ventura; Curso de geometria. ISBN: 972-662-591-2

Complementary Bibliography

Coxeter H. S. M.; Geometry revisited
Moise Edwin E.; Elementary geometry from an advanced standpoint

Teaching methods and learning activities

Lectures and classes: The contents of the syllabus are presented in the lectures, where examples are given to illustrate the concepts. There are also practical lessons, where exercises and related problems are solved. Both in lectures and practical classes, dynamical geometry software will be used for the constructions with rule and compass, illustrating the theorems, and in some cases, explaining them to a certain extent.

Software

Compass and Ruler - Construct and Rule (http://car.rene-grothmann.de/doc_en/)

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Calculation formula of final grade

Final exam, where two groups of questions, worth 4/20 each, may be replaced, by the student's choice, by the results of two essays.