Two-dimensional hypercomplex numbers and related trigonometries and geometries

All the commutative hypercomplex number systems can be associated with a geometry. In two dimensions, by analogy with complex numbers, a general system of hypercomplex numbers can be introduced and can be associated with plane Euclidean and pseudo-Euclidean (space-time) geometries. In this paper we show how these systems of hypercomplex numbers allow to generalise some well known theorems of the Euclidean geometry relative to the circle and to extend them to ellipses and to hyperbolas. We also demonstrate in an unusual algebraic way the Hero formula and Pytaghoras theorem, and show that these theorems hold for the generalised Euclidean and pseudo-Euclidean plane geometries.