Basic Geometric Properties of Triangles

In this work, we define angles between vectors and between three points. Building on this, we prove basic geometric properties of triangles, such as the Isosceles Triangle Theorem, the Law of Sines and the Law of Cosines, that the sum of the angles of a triangle is π, and the congruence theorems for triangles. The definitions and proofs were developed following those by John Harrison in HOL Light. However, due to Isabelle’s type class system, all definitions and theorems in the Isabelle formalisation hold for all real inner product spaces.