﻿ Between: Point C on a line is between points A and B iff AC+CB=AB.

# Between

Pronunciation: /bɪˈtwin/ ?

Point B is between two other points A and C if it is on line segment AC[1]. In his book "The Foundations of Geometry"[2], German mathematician David Hilbert organized the concept of between into five axioms called the Axioms of Order.

Axioms of Order
AxiomManipulativeDiscussion
II,1
 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Manipulative 1: Axiom of order II,1. Click on the blue points in the manipulative to change the figure.
"If A, B, and C are points of a straight line and B lies between A and C, then B lies also between C and A."[2]
This tells us that if point B is between A and C, then it is also between points C and A.
II,2
 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Manipulative 2: Axiom of order II,2. Created with GeoGebra.
"If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D."[2]
This axiom establishes that between any two points on a line, there exists another point. A consequence of this axiom is that an infinite number of points lie between any two points on a line.
II,3
 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Manipulative 3: Axiom of order II,3. Created with GeoGebra.
"Of any three points situated on a straight line, there is always one and only one that lies between the other two."[2]
This says that if point B lies between points A and C then point A can not lie between points B and C.
II,4
 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Manipulative 4: Axiom of order II,4. Created with GeoGebra.
"Any four points A, B, C, D of a straight line can always be arranged so that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D."[2]
This axiom establishes that point on a line can always be put in order.
II,5
 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Manipulative 5: Axiom of order II,5. Created with GeoGebra.
"Let A, B, C be points not lying in the same straight line and let a be a straight line lying in the plane ABC and not passing through any of the points A, B, C. Then, if a straight line passes through a point of segment AB, it will always pass through either a point of the segment BC or a point of the segment AC."[2]
This axiom establishes that a straight line that intersects one side of a triangle, must also intersect a second side of the same triangle.

• McAdams, David E.. Segment Addition Postulate. allmathwords.org. Life is a Story Problem LLC. 2010-03-16. http://www.allmathwords.org/article.aspx?lang=en&id=Segment Addition Postulate.

Between. 2010-01-08. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/b/between.html.

### Revision History

2010-01-08: Rewrote article from point of view of "Foundations of Modern Geometry". (McAdams, David.)