Betweenness of Points By definition, a point B is between two other points A and C if all three points are collinear and AB +BC =
Theorem Given three distinct points on a line in a metric geometry, one and only one of them is between the other two Proof Immediate consequence of the
12 nov 2015 · Def: Betweenness of Points?–?A point is between two other points on the same line iff its coordinate is between their coordinates ??(More
When we say equality we mean sameness, not congruence or “same size” For example, when the objects are sets (such as line segments, circles, rays), equality
Suppose C is a nonempty family of convex sets of points Then ?C is convex Proof This is obvious Rays Definition Suppose o and a are distinct points
For example, in geometry, the most common undefined terms are “point” and “line ” Definition 3 15 (Betweenness) For any three points A, B and C,
A system satisfying the incidence and betweenness axioms is an ordered For example, lines and planes in Euclidean geometry (affine subspaces) are cosets
necessity of stating any axioms that would give him betweenness; he vation for the definition, the geometry is then entirely excluded from
the relation of betweenness in linear order such cannot be the case since four been made to make the number of points in each example the least possible
(Cont) Any geometric linear ordering of a line is complete Halfspaces A lot of what we will do here will bear a resemblance to what we did with rays Definition
We can use distance to define betweenness, as follows: Definition 4 1 A - B - C if and only if A, B, C are three distinct collinear points and AB + BC = AC