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.