Birkhoff's axioms

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

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

Postulate III: Postulate of Angle Measure. A set of rays 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.

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