At the end of this course the student will have acquired knowledge and skills that will enable him:
To know and understand some of the principal results on the Euclidean
To develop skills to solve geometric problems and no systematic ones
To be able to visualize in the space

General characterization





Responsible teacher

Ana Cristina Malheiro Casimiro


Weekly - 4

Total - 70

Teaching language



Knowledge of the contents of Linear Algebra and Analytic Geometry I.


"Foundations of Geometry" (2nd edition), Gerard A. Venema, 2006, Pearson

“Curso de Geometria”, Paulo Ventura Araújo, Trajectos Ciência, 1998, Gradiva

"A course in Modern Geometries" (2nd edition), Judith N. Cederberg, 2001, Undergraduate Texts in Mathematics, Springer

“Geometry and symmetry”, L. Christine Kinsey, T. E. Moore, E. Prassidis 2011, John Wiley and Sons

“Geometry, ancient and modern”, J.R. Silvester, 2001, Oxford Univ. Press

“Axiomatic Geometry”, John M. Lee, 2013, American Mathematical Society

“Continuous Symmetry From Euclid to Klein”, W. Barker e R. Howe, 2007, American Mathematical Society

Teaching method

There are classes in which theory is lectured and illustrated by examples. There are also problem solving sessions. Some exercises are left to the students to be solved on their own as part of their learning process. Students can ask questions during the classes, in weekly scheduled sessions or in special sessions accorded directly with the professor. There are two mid term tests that can substitute the final exam in case of approval. Otherwise the student must pass the final exam. In order to be evaluated, students must attend, at least, 2/3 of the lectures and 2/3 of the problem solving classes.

Evaluation method

Geometry - 2022/2023

Evaluation Rules

1. Continuous evaluation

During the semester two tests will be carried out with a duration of 1 hour 30 minutes. Each test is rated up between 0 and 20 values. The classification of continuous evaluation (AC) is obtained by the mean of the tests classification.

The student is approved in the course if AC is greater than or equal to 9.5 values. 

2. Exam

At the date and time scheduled for the Exam any student enrolled in the course who has not obtained approval in the Continuous Evaluation can take the exam for 3 hours. If its classification is greater than or equal to 9.5, the student is approved with this classification.

3. Grade improvement

Students have the right to improve grade by enrollment within the established deadlines, at the time of the Exam. In this case, they may take the 3-hour Exam as described in the previous paragraph. 


4. Final considerations

In all that this Regulation is missing, the FCT-UNL General Regulations are valid.

Subject matter

1. Axiomatic geometry: introduction, incidence geometry, axioms for plane geometry, angles, triangles, models of neutral geometry (cartesian model, Poincaré disk model), parallel axiom (equivalence with the sum of the inner angles of a triangle is 180º).

2. Straightedge and compass constructions: basic constructions, construction of regular polygons, impossible constructions.

3. Groups of transformations in the plane and in the Euclidean space: affinities, similarities and isometries and its classification

4. Non Euclidean geometries: models of the projective or elliptic or hyperbolic planes (to choose one of the 3 geometries)