Equidistance and Betweenness in Euclidean Plane Geometry

Tarski axiomatized Euclidean plane geometry in first-order logic using two primitive relations: " lies between and ," written as , and " is as distant from as is from ," written as . This Demonstration shows that in plane Euclidean geometry, the relation is definable using relation , but the opposite is not true.
The first step is to define
.
Using that relation, we define
.
To show that cannot be defined using , two models must be given that preserve the relation but not the relation . The first model is the Cartesian product of real numbers. The second is obtained by using the linear transformation . This transformation preserves lines. We have , but is not true.

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DETAILS

Notation and definitions are from [2].
References
[1] A. Tarski, "What Is Elementary Geography?," The Axiomatic Method: With Special Reference to Geometry and Physics (L. Henkin, P. Suppes and A. Tarski, eds.), Amsterdam: North-Holland, 1959 pp. 16–29.
[2] A. Tarski and S. Givant, "Tarski's System of Geometry," Bulletin of Symbolic Logic, 5(2), 1999 pp. 175–214. www.math.ucla.edu/~asl/bsl/0502/0502-002.ps.
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