Subject: Euclidean Geometry

Scientific Area:



80 Hours

Number of ECTS:

7,5 ECTS



Overall objectives:

1 - Knowing a little of the historical origin of geometry in particular how the geometry was discussed in ancient Greece.
2 - Ability to make geometric reasoning and deductions.
3 - Knowing the most important classical geometric constructions.
4 - Ability to reason in more abstract geometric form.
5 - Meet other geometries and identify their properties.
6 - Consolidation of the use of coordinate systems on an affine n-dimensional space.
7 - Characterization of affine sub-spaces (lines, planes, hyperplanes, etc.) through their equations.
8 - Identification of the relative position of subspaces defined by linear equations.
9 - Know and be able to deduce geometric properties of objects on the Euclidean plane and in space.
10 - Ability to apply knowledge of geometry to calculate the relative positions, distances and angles of geometric objects.


1 - History of geometry. Euclid's Elements. The problem of parallels.
2 - Incidence. Non-Euclidean and Euclidean distances. Segments and half-lines. Convex sets. Unbundling. Angles and triangles. Angular Measurement and triange congruences. Axiom of parallelism and their equivalents. Paraelel projections and similar triangles. Hyperbolic and projective planes and their properties.
3 - Referencials and coordinate systems. Points, lines and planes as a solution of systems of linear equations in two and three variables. Solving a linear system of equations as intersection of hyperplanes on an euclidean n-dimensional spaces. Affine spaces. Parallelism.
4 - Inner products.Ortgho normal basis and Gram-Schmidt orthogonalization method. Euclidean spaces. Orthogonal complement of a subspace. Orthogonal projections. Distance between points and lines between points and planes, between two lines, between a line and a plane and between two planes. Generalization to the distance between two subspaces of an Euclidean space. Angle between two subspaces.


Oliveira, A.J. Franco de , 1995 , Geometria euclidiana , Universidade Aberta
Monteiro, António , 2001 , Álgebra linear e geometria analítica , McGraw-Hill

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Theoretical classes are expository. Several examples are presented to facilitate the understanding of concepts and serve as motivation for students. Students are provided with a list of problems and exercises that they should try to solve outside the classroom. In theoretical-practical classes, these problems are discussed and solved to overcome any difficulties that students may have. Two tests are performed. First test about Synthetic Geometry and the second test about Analytical Geometry. Each test has a weight of 50% in the final grade.