Birkhoff's axioms

Definition

Let $\Pi$ be a set of points with a family of subsets called lines and a distance function $d \colon \Pi \times \Pi \to \R$. For a point $O$, the rays from $O$ carry a measure: $\angle AOB$ denotes the angle of the rays $OA$ and $OB$. Birkhoff's axioms are the following four postulates.

1. Line measure: the points of each line $\ell$ can be put in bijection with the real numbers, $A \mapsto x_A$, so that $\abs{x_A - x_B} = d(A, B)$ for all $A, B \in \ell$.
2. Point and line: through two distinct points there is exactly one line.
3. Angle measure: the rays from each point $O$ can be put in bijection with $\R / 2\pi\Z$. Then $\angle AOB$ is the difference of the numbers of $OA$ and $OB$. Moreover, as $B$ moves continuously on a line not through $O$, the number of the ray $OB$ varies continuously.
4. Similarity: let $ABC$ and $A'B'C'$ be triangles and $k > 0$ such that $d(A', B') = k\, d(A, B)$, $d(A', C') = k\, d(A, C)$ and $\angle B'A'C' = \pm \angle BAC$. Then $d(B', C') = k\, d(B, C)$, $\angle C'B'A' = \pm \angle CBA$ and $\angle A'C'B' = \pm \angle ACB$.