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.
 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.