Birkhoff Axioms

Overview

Birkhoff presented an axiomatization of Euclidean geometry quite different in spirit from Hilbert’s or Euclid’s, offering a concise yet powerful system. As with earlier masters, terms such as point, line, distance, and angle are accepted and used without definition.

Axioms ¹

$d$ is a distance.

Definitions

If $d (A,B) + d (B,C) = d (A,C)$, then point $B$ is said to be between points $A$ and $C$.
Points $A$ and $C$ together with all points $B$ between them form the line segment $\overline{AC}$.
An endpoint $O$ and a half-line $m '$ are, for two points $O$ and $A \ne O$ on the line $m$, defined as the set of all points $A '$ such that $O$ is not between $A$ and $A '$.
If two distinct lines have no point in common, they are parallel. A line is always considered parallel to itself.
Let two half-lines $m$ and $n$ passing through $O$ be given. If $\angle{mOn} = \pm \pi$, they form a straight angle; if $\angle{mOn} = \pm \pi / 2$, they form a right angle, and $m$ is said to be perpendicular to $n$.
Given three distinct points $A$, $B$, $C$, the three segments $\overline{AB}$, $\overline{BC}$, $\overline{CA}$ form a triangle $\triangle{ABC}$ with sides $\overline{AB}$, $\overline{BC}$, $\overline{CA}$ and vertices $A$, $B$, $C$. If the three points lie on a single line, $\triangle{ABC}$ is said to be degenerate.
Two figures are similar if there exists a one-to-one correspondence between their points such that corresponding distances are proportional and corresponding angles are equal. Two geometric figures are congruent if they are similar with the scale factor $k = 1$.

Postulates

Postulate of line measurement: The points $A, B, \cdots$ of any line $m$ can be put into one-to-one correspondence with real numbers $r_{A}, r_{B}, \cdots$ so that $\left| r_{B} - r_{A} \right| = d (A,B)$.
Point-line postulate: There exists a unique line $m$ containing any two distinct points $P, Q$.
Postulate of angle measurement: The half-lines $m , n$ through any point $O$ can be put into one-to-one correspondence with real numbers $a \pmod{2 \pi}$ so that $A \ne O$ and $B \ne O$ correspond to points of $m$ and $n$ respectively, and $\left( a_{n} - a_{m} \right) \pmod{2 \pi}$ is the measure of $\angle{AOB}$.
Postulate of similarity: For triangles $\triangle{ABC}$ and $\triangle{A ' B ' C'}$ and any positive constant $k$, if $d \left( A ' , B ' \right) = k d \left( A , B \right)$ and $d \left( A ' , C ' \right) = k d \left( A , B \right)$ and $\angle{BAC} = \pm \angle{B ' A ' C '}$, then $d \left( B ' , C ' \right) = k d \left( B , C \right)$ and $\angle{A ' B ' C '} = \pm \angle{ABC}$ and $\angle{A ' C ' B '} = \pm \angle{ACB}$.