One point B can lie between two other points A and C {intermediacy}: ABC. All intermediate points B make a segment. All intermediate points B, plus points A and C, make an interval. All points past A in direction AC make a ray. An interval and its two rays make a line. Two rays with a common point make an angle.
Non-metric geometry {ordered geometry} can use intermediacy. Ordered geometry can be the basis of affine geometry and its sub-geometries: Euclidean geometry (parabolic geometry), absolute geometry, and hyperbolic geometry, because straight lines, parabolas, and hyperbolas are open figures and have between-ness. Ordered geometry cannot be the basis of projective geometry, because circles and ellipses are closed figures and so do not have between-ness.
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Date Modified: 2022.0224