Orthogonality as well as Equidistance Can Be Used as the Sole Primitive Notion for Euclidean Geometry

Pieri has shown that the ternary relation of a point equally distant from two other points and (in symbols, ) can be used as the primitive foundation of Euclidean geometry of two or more dimensions [1].

This Demonstration shows that the relation that is the midpoint of can be defined using the relation that , and form a triangle with a right angle at . The definition is

Pieri's relation can be defined by

.

Thus, Euclidean geometry can also use the relation as a simple foundation.

Robinson's definitions of geometric relations using Pieri's relation [3, pp. 71–72] are:

Here, means is between and ; means , and are collinear; and means and are symmetric about (i.e. that is the midpoint of ).

References

[1] M. Pieri, "La Geometria Elementare istituita sulle nozioni di punto e sfera," Memorie di matematica e di fisica della Societ`a italiana delle Scienze, ser. 3(15), 1908 pp. 345–450.

[2] H. L. Royden, "Remarks on Primitive Notions for Elementary Euclidean and Non-Euclidean Plane Geometry," in Studies in Logic and the Foundations of Mathematics, Vol. 27, Amsterdam: North-Holland Publishing Company, 1959 pp. 86–96.

[3] R. M. Robinson, "Binary Relations as Primitive Notions in Elementary Geometry: The Axiomatic Method with Special Reference to Geometry and Physics," in Proceedings of an International Symposium Held at the University of California, Berkeley, December 26, 1957–January 4, 1958, Amsterdam: North-Holland Publishing Company, 1959 pp. 68–85. doi:10.1017/S0022481200092690.