linguistic:paraphrasing
Congruence and betweenness

Three points equidistant from two distinct points form a line. Without this axiom, the theory could be modeled by three-dimensional or higher-dimensional space.

Three variants of this axiom can be given, labeled A, B and C below. They are equivalent to each other given the remaining Tarski's axioms, and indeed equivalent to Euclid's parallel postulate.

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.

Given any triangle, there exists a circle that includes all of its vertices.

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.

Each variant has an advantage over the others:

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.

This is equivalent to the side-angle-side rule for determining that two triangles are congruent; if the angles uxz and u'x'z' are congruent (there exist congruent triangles xuz and x'u'z'), and the two pairs of incident sides are congruent (xu ≡ x'u' and xz ≡ x'z'), then the remaining pair of sides is also congruent (uz ≡ u'z').

For any point y, it is possible to draw in any direction (determined by x) a line congruent to any segment ab.