Pieri's Ternary Relation and Euclidean Geometry

M. Pieri showed that the ternary relation of a point being equally distant from two other points (in symbols, ) can be used as the only primitive notion of Euclidean geometry of two or more dimensions [1]. This Demonstration shows Robinson's definition of the quartic relation
in terms of Pieri's relation. This relation can be used to define relations that a point is between and ; that , , are collinear; and that is the midpoint of .
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Here are Robinson's definitions [2, pp. 71–72]:
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] 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. doi:10.1017/S0022481200092690.
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