The Fundamental Theorems of Elementary Geometry. an Axiomatic Analysis

Introduction. The present investigation is concerned with an axiomatic analysis of the four fundamental theorems of Euclidean geometry which assert that each of the following triplets of lines connected with a triangle is copunctual: the medians, the altitudes, the perpendicular bisectors, and the bisectors of the angles. The general framework for our discussion will be provided by an affine plane (which is obtained from a projective plane by deleting a line and its points). But in order to enunciate these theorems we have to add to the concepts provided by affine geometry two further relations: the relation of the midpoint and that of orthogonality. There exists one important difference between these two relations. If one subjects the midpoint relation to some obvious and formal restrictions, then there exists at most one such relation. But there exist always different orthogonality relations, since affine transformations transform one such relation into a different one; and even if one considers such orthogonality relations as not essentially different and considers only orthogonality relations meeting quite a fair amount of requirements, then uniqueness will be an exceptional case. The existence of a midpoint relation is equivalent to the closure of certain configurations, to the existence of sufficiently many reflections in points, and to the following algebraic criterion : the plane under consideration is the plane over a right distributive Cartesian number system of characteristic different from 2 (using a concept introduced in an earlier paper(1)). If the characteristic is 3, then the medians of a triangle are parallel, otherwise they are copunctual. Given a midpoint relation and an orthogonality relation meeting the obvious and formal requirements, then the theorem of the altitudes and that of the perpendicular bisectors of a triangle are equivalent; and they are both equivalent to the fact that the plane under consideration is the plane over an ordinary commutative field of characteristic different from 2 and that orthogonality may be defined in terms of a quadratic form y2 — cx2, c^O. The relation between these theorems and the theorem of the bisectors of the angles is not as clearcut. For, the latter theorem is a consequence of the former ones; but the converse can be obtained only by adding two further statements concerning the existence of bisectors of angles, which does not seem to be assured by