Segment, ray and comparing segments

Axiom: of any three points on a line, exactly one lies between the other two. If we mark two points on a line, those two points together with every point between them form a segment: the outer points are its endpoints and the rest are its interior points. If we choose one point on a line and take the part going from it in one direction only (including the point), we get a ray; the chosen point is the origin of the ray. When a ray is written the origin comes first: ray OA. If points A and B lie in the same half-plane of a line, segment AB does not cross the line; if they lie in different half-planes it always crosses it.