Article : Half-plane geometry - Liceo Scientifico R. Bruni (Padova)

This article studies the geometry of Poincaré's half-plane, i.e. the half-plane limited by a straight h where the lines are half-circles centered on h, and where the axioms of classical Euclidean geometry hold except for the axiom of parallel lines. Lacking the notion of distance, congruence between segments is introduced by constructing shifts from one line to another. After that, it is shown that the sum of the angles of a triangle is always less than 180 degrees and then the properties of many polygons are investigated.