Betweenness axiom: Betweenness for points on a line corresponds to betweenness for rays from a point.
Definition: If A and B are distinct points, the point C is between A and B if C is not equal to A or B, and C lies on the segment with endpoints A and B.
Definition: If A, B, and C are different from O, and @(A,O,B) ≠ 180, we say that the ray OC is between the ray OA and the ray OB if
0 < @'(A,O,C) < @'(A,O,B) or 0 > @'(A,O,C) > @'(A,O,B),
where @' denotes the proper angle measure.
Betweenness Axiom
Let A, O, B be three different points such that a(A,O,B) ≠ 180.
1. If C is between A and B, then the ray OC is between the ray OA and the ray OB.
2. If D is not O, and the ray OD is between the ray OA and the ray OB, then the ray OD intersects the segment AB .
Points on the same side of a line
This concept is closely related to betweenness.
Definition: Let A, B, C, D be points such that A ≠ B, C ≠ D, and the points C and D are not on the line through A and B. The points C and D are on the same side of the line AB if the proper angle measures @'(A,B,C) and @'(A,B,D) have the same sign.
Concepts like betweenness and the sides of a line are part of the "geometry of position," the precurser of the modern subject of topology.
We follow Birkhoff in using angle measure @(A,O,B) to define concepts from the geometry of position. If only absolute angle measure a(A,O,B) is used, then these concepts are usually taken for granted and considered as "obvious" from diagrams, just as in Euclid's Elements. This is what is done in almost all high school geometry books -- including Birkhoff's! And this is the viewpoint we will take in practice in this course.
Hilbert's approach was to develop properties of betweenness axiomatically. This leads to a long list of subtle abstract axioms. The SMSG axioms and their modern descendants are a mixture of Hilbert's and Birkhoff's viewpoints. (See my history of axioms for plane geometry.)