Birkhoff's axioms

In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry. Other often-used axiomizations of plane geometry are Hilbert's axioms and Tarski's axioms.

Postulates

Postulate I: Postulate of Line Measure. A set of points {A, B...} on any line can be put into a 1:1 correspondence with the real numbers {a, b...} so that |b-a| = d(A,B) for all points A and B.

Postulate II: Point-Line Postulate. There is one and only one line, l, that contains any two given distinct points P and Q.

Postulate III: Postulate of Angle Measure. A set of 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 (not equal to O) of l and m, respectively, the difference a_m - a_l (mod 2π) of the numbers associated with the lines l and m is $angle$AOB.

Postulate IV: Postulate of Similarity. Given two triangles ABC and A'B'C' and some constant k>0, d(A', B') = kd(A, B), d(A', C')=kd(A, C) and $angle$B'A'C'=±$angle$BAC, then d(B', C')=kd(B,C), $angle$C'B'A'=±$angle$CBA, and $angle$A'C'B'=±$angle$ACB

References

* Birkhoff, G. D., 1932, "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)," "Annals of Mathematics 33".