# Betweenness of Points: Definition & Problems

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• 0:01 What is Betweeness?
• 0:43 What Does This Mean?
• 1:30 Theorem of Betweeness
• 2:14 Example
• 3:34 Lesson Summary

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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

While it sounds very basic, the idea of betweenness has some pretty big implications for math. In this lesson, see not only why it matters, but also how we can use it to help prove basic geometrical problems.

## What is Betweenness

While it sounds unbelievable, one of the central theories of geometry wasn't established until the 20th century. And it is so simple that it only takes a basic understanding of math to understand. Don't you wish you had been the one to notice it? There was a relatively recent development, the idea of betweenness, is central to our idea of understanding of math in the real world. In short, it means that if a point B is between points A and C, the length of line AB plus the length of line BC will equal the length of line AC. With it, much of geometry makes sense. Without it, all of geometry falls apart.

## What Does This Mean?

Let's back up for a second and make sure that we understand the point of betweenness in the first place. You can draw a straight line between any two points no matter where they are. However, once you add a third point, there is the chance that point is an outlier, and no straight line can connect all three points. By establishing betweenness, we can continue to use many of the advantages of having two points.

Think about a typical polygon, say a parallelogram for example. If we can establish that points on a line segment that forms one of the sides are between each other, then we can say that the lines are straight. Without that knowledge, we suddenly end up with parallelograms that have curves for sides. As you can imagine, this makes formulas, like base x height to find the area, very inaccurate depending on the shape of the curves.

## Theorem of Betweenness

Alongside these fairly obvious benefits of betweeneness there is also the theorem of betweenness that helps us ascertain whether or not the points on a line actually meet the requirements for betweenness. All in all, it's pretty simple, even though it requires some setup.

Let's say you have a line segment with three points, A, B, and C. The line is straight and all three points are on it in the sequence of A, B, and C. The theorem of betweenness tells us that the length of AC is the sum of AB and BC. Now this is only true if B is between A and C, otherwise it would be false. By the same token, if you don't have the length of AB but know BC and AC, you can then find the total length of AB. For thinking that sounds a little circular, don't worry, I've got an example that will help make some sense of it all.

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