# Affine and Euclidean Geometry: Course Outline

In mathematics, affine geometry is what remains of Euclidean geometry when not using (mathematicians often say "when forgetting") the metric notions of distance and angle.

## Course Outline

Vector spaces and affine geometry:

Collinearity of three points, ratio AB/BC. Linear combinations and linear dependent set versus affine combinations and affine dependent sets. Classical theorems in affine geometry:  Thales,  Menelaus, Ceva,  Desargues.

Affine  subspaces, affine maps. Dimension of a linear subspace and of an affine subspace.

Euclidean  geometry:

Scalar  product,  Cauchy-Schwartz inequality: norm of a vector, distance between two points, angles between two non-zero vectors. Pythagoras theorem, parallelogram law, cosine and sine rules. Elementary geometric loci.

Orthogonal  transformations:

Isometries  of   plane  (four  types), Isometries of space (six types).Orthogonal bases.

Platonic   polyhedra

Euler   theorem   on   finite   planar   graphs. Classification of  regular  polyhedra  in  space.

Isometries  of  regular polygons and regular polyhedra.