Details

If A is between B and C, A is not necessarily adjacent to B or C.

M. Piery has shown that a ternary relation, that of a point being equally distant from two other points, can be used as the only primitive notion of Euclidean geometry of two or more dimensions [1, p. 68].

Reference

[1] 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.