An application of Pappus' Involution Theorem in euclidean and non-euclidean geometry

Pappus’ Involution Theorem is a powerful tool for proving theorems about non-euclidean triangles and generalized triangles in Cayley-Klein models. Its power is illustrated by proving with it some theorems about euclidean and non-euclidean polygons of different types. A $n$-dimensional euclidean version of these theorems is stated too.