Congruence and betweenness
Lower Dimension
In short, there exist three noncollinear points, and any model of these axioms must have dimension > 1.
Upper dimension
Upper Dimension
Three points equidistant from two distinct points form a line. Hence any model of these axioms must have dimension < 3.
Axiom of Euclid
Each of the three variants of this axiom, all equivalent over the remaining Tarski's axioms to Euclid's parallel postulate, has an advantage over the others:
A dispenses with existential quantifiers;
B has the fewest variables and atomic sentences;
C requires but one primitive notion, betweenness. This variant is the usual one given in the literature.
A: 
Let a line segment join the midpoint of two sides of a given triangle. That line segment will be half as long as the third side. This is equivalent to the interior angles of any triangle summing to two right angles.
B: 
Given any triangle, there exists a circle that includes all of its vertices.
Axiom of Euclid: С
C: 
Given any angle and any point v in its interior, there exists a line segment including v, with an endpoint on each side of the angle.
Five Segment
Five segment
Begin with two triangles, xuz and x'u'z'. Draw the line segments yu and y'u', connecting a vertex of each triangle to a point on the side opposite to the vertex. The result is two divided triangles, each made up of five segments. If four segments of one triangle are each congruent to a segment in the other triangle, then the fifth segments in both triangles must be congruent.
Segment Construction
Given any two line segments, the second can be "extended" by a line segment congruent to the first.
Such economy of primitive and defined notions means that Tarski's system is not very convenient for doing Euclidian geometry.
Rather, Tarski designed his system to facilitate its analysis via the tools of mathematical logic, i.e., to facilitate deriving its metamathematical properties.
Tarski's
system has the unusual property that all sentences can be written in
universal-existential form, a special case of the prenex
normal form.
This form has all universal
quantifiers preceding
any existential
quantifiers,
so that all sentences can be recast in the form 
This fact allowed Tarski to prove that Euclidean geometry is decidable: there exists an algorithm which can determine the truth or falsity of any sentence.
Tarski's axiomatization is also complete. This does not contradict Gödel's first incompleteness theorem, because Tarski's theory lacks the expressive power needed to interpret Robinson arithmetic (Franzén 2005, pp. 25–26).
Упражнения.
Сформулировать в системе Тарского 5 геометрических теорем на Ваш выбор,
в частности,
« В любом треугольнике медианы пересекаются в одной точке и делятся точкой
пересечения на отрезки, длины которых относятся как 2:1».