Birkhoff’s Axioms for Euclidean Geometry

Undefined Terms

• points A, B, ...
• line - A set of points.
• distance - The distance between any two points A and B is a non-negative real number d(A,B) such that d(A,B) = d(B,A)
• angle - An angle is formed by three ordered points A, O, B (A ≠ O, B ≠ O) where m(∠AOB) is a real number (mod 2π). The point O is called the vertex of the angle.

Definitions

Between

If A, B and C are distinct points, we say that B is between points A and C if and only if d(A,B) + d(B,C) = d(A,C).

Line Segment

The points A and C together with all points B between A and C form the line segment AC.

Half-Line; Endpoint

The half-line m' with endpoint O is defined by two points O, A in line m (A ≠ O) as the set of all points A' of m such that O is not between A and A'.

Parallel

If two distinct lines have no points in common they are parallel. A line is always regarded as parallel to itself.

Straight Angle; Right Angle; Perpendicular

Two half-lines m, n through O are said to form a straight angle if m(∠mOn) = π. Two half-lines m, n through O are said to form a right angle if m(∠mOn) = π/2, in which case we say that m is perpendicular to n.

Triangle; Vertices; Degenerate Triangle

If A, B, C are three distinct points the three segments AB, BC, CA are said to form a triangle with sides AB, BC, CA and vertices A, B, C. If A, B and C are collinear then triangle ΔABC is said to be degenerate.

Similar; Congruent

Any two geometric figures are similar if there exists a one-to-one correspondence between the points of the two figures such that all corresponding distances are in proportion and corresponding angles have equal measures (except, perhaps, for their sign). Any two geometric figures are congruent if they are similar with a constant of proportionality, k=1.

Postulates

Postulate I.

Postulate of Line Measure. The points A, B, ..., of any line can be put into 1-to-1 correspondence with the real numbers x so that
|x_a - x_b| = d(A,B) for all points A and B.

Postulate II.

Point-line Postulate. One and only one line, l, contains any two distinct points P and Q.

Postulate III.

Postulate of Angle Measure. The half-lines (or rays) l, m, n, ..., through any point O can be put into 1-1 correspondence with the real numbers a (mod 2π) so that if A and B are points (other than O) of l and m, respectively, the difference a_m - a_l (mod 2π) of the numbers associated with lines l and m is m(∠AOB).

Postulate IV.

Postulate of Similarity. If in two triangles ΔABC and ΔA'B'C' and for some constant k>0, d(A',B')=k d(A,B), d(A',C')=k d(A,C), and
m(∠B'A'C')= m(∠BAC), then also d(B',C')=k d(B,C), m(∠C'B'A')= m(∠CBA), and m(∠A'C'B')= m(∠ACB).

Sources:

1. A Set of Postulates for Plane Geometry (Based on Scale and Protractors), G. D. Birkhoff, Annals of Mathematics, 33, 1932
2. Roads to Geometry, E. Wallace & S. West, Prentice-Hall (Sect 2.5 & App. C)