Geometry

Code:

M1028

Acronym:

M1028

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 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:M	110	Official Study Plan	1	-	6	48	162

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. 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. Possible introduction to the concept of inversion. 

Mandatory literature

Peter Gothen e António Guedes de Oliveira; Geometria Euclidiana - com construções interativas, U.Porto Press, 2021. ISBN: 9789897462931
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

Comments from the literature

I provide notes

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. 

Software

Compass and Ruler - Construct and Rule (http://car.rene-grothmann.de/doc_en/)
Geogebra (https://www.geogebra.org/)

Evaluation Type

Distributed evaluation without final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	114,00
Frequência das aulas	48,00
Total:	162,00

Eligibility for exams

No frequency requirements are required.

Calculation formula of final grade

• Regular season: the first part will be assessed in the middle of the semester, on a date to be announced, and the second part on the date scheduled for the regular season examination (there will be no final exam). For students attending TP classes (will have to attend 9 TP classes at a minimum), there may be small quizzes in class that may be used to replace certain questions in the regular season tests.


• Second evaluation season: Students can choose to be evaluated only to one of the parts, getting for the other part the grade obtained during the regular season. If students choose to be evaluated to both parts, the final grade of each part will be calculated as follows (CN - rank of that part in the regular season; CR - rating that part in the second evaluation season):




• If CR>=0.8*CN, the final grade of that part will be maximum{CN,CR};


• If CR<0.8*CN, the final grade of that part will be mean{CN,CR}.

Special assessment (TE, DA, ...)

Either oral or written examination.

Classification improvement

• Students aproved last year who want to improve their classification in the first evaluation period will have to take both tests during the semestre.


• Students who want to improve their classification at the at the second evaluation season, may choose to improve only the classification of one of the tests, keeping the classification of the first evaluation season in the other test. For this, they will have to inform, in advance, by email, the regent of the UC.


• For students who want to improve the classifications on both tests, the classifications of the first evaluation season will not be taken into account in the calculation of the classification of the improvement exam.

Observations

Any student can be called to an extra test (oral or written) to clarify any doubts that may have arisen.