In Euclidean geometry, two or more points on a line that are close to or far from each other are said to be collinear. The word "collinear" is derived from **the Latin word columella**, meaning "a supporting beam." Thus, three things that are colinear are lying along the same beam.

Two points are called collinear if they lie on the same line but not all of them have to be marked as such. If necessary, you can mark one point by putting a dot next to it or even giving it a number. There are several ways of representing **four or more points** that are collinear. You can show all of them on one diagram by labeling each one of them with a distinct letter, like so: a b c D. Or, if you want to save space, you can put a symbol for **a straight line** between any two of them that are close together and leave out the rest, like this: a - b - C. A special case occurs when there are only three points involved, because in this case there is only one way they can be collinear and that's if they are on the same line.

Here is an example of four points that are collinear: 4, 7, 10, 13.

Collinear points are points that are parallel to a line. Any two points that are always collinear because a straight line can always join them. These points can be on the same object or different objects.

For example, if you look at **this picture** of the Collinwood Cemetery in Columbus, Ohio, you can see that many monuments are located along a single path. These monuments are all collinearly aligned because the path they are on is shaped like a line from start to finish. You can also see that some of the stones are larger than others; these represent differences in age and status among the people buried there. The oldest person buried there was born in 1807 and lived for **114 years** and 6 months. The newest person buried there was born in 2000 and died just eight days later.

All seven people buried here were leaders who helped shape America into what it is today. They all had **strong beliefs** about how society should work and they all tried to make their world better by helping other people. No one is actually buried here; this is only a monument to show where they are now because none of them were alive when they were put on display.

The first thing you will notice about this cemetery is the large number of monuments that mark the graves of **each individual**.

Three or more points,... are said to be collinear if they lie on a single straight line. A line on which points lie, especially if it is related to a geometric figure such as a triangle, is sometimes called an axis. Two points are trivially collinear since two points determine a line. Any three non-collinear points define a plane. A fourth point that does not lie in this plane but that is not equal to one of the first three points will also not lie in the plane.

A fifth point that lies outside this plane is called a collinear point. These can occur in many configurations including inside, outside, above, and below the plane defined by the first four points. There are five types of collinear configurations: opposite, adjacent, parallel, perpendicular, and mixed.

Opposite means that the fifth point is on **the other side** of the first point from the second point. For example, if you were to draw a diagram of **the following configuration**, you would say that p5 is opposite p1. Adjacent means that the fifth point is between the first two points (not including their relationship to each other). So for example, if you were to draw a diagram of the following configuration, you would say that p5 is adjacent to p1 and p4. Parallel means that the lines formed by **each pair** of points intersect at a single point.

Collinear points are three or more points on the same line. For instance, consider **the points A**, B, and C on the line m. They are parallel. On the line n, the points D, B, and E are located. These points are also collinear.

If you draw three lines through these points that don't intersect, each of them will pass through exactly two of them. This means that there are always pairs of points that do not lie on the same line.

According to this definition, every pair of lines in 3D space determines a unique point (the intersection of both lines). So, in order for these definitions to make sense, you need at least three dimensions. In four dimensions, you would have a volume of space where any two lines would define a point - but since we're only counting lengths here, that volume must be infinite.

In general, if you have d-dimensional space, then there are d+1-dimensional hyperplanes that can be defined by **any set** of **d lines**. Each of **these hyperplanes** will contain d points, so they will all be different. There cannot be more than d+1-dimensional surfaces in d-space, because if there were, they would overlap.

Because we can draw a distinct (one) line across two locations, they are always collinear. If three points are on the same line, they are collinear. Points A, B, and C are not converging. We can construct a line between A and B, A and C, and B and C, but not between all three places. Therefore, the points are collinear.

Collinear points share the same property as parallel lines: They cannot be used to find the distance between two locations. However, they do provide a way to compare distances within the same location. If one point is closer to **another point** than it is to some other point, then they are collinear.

Closeness in space can also be defined in terms of angles. If two locations are close together, their corresponding angles will be small. For example, if there is some angle between **0 and 10 degrees**, then they are collinear. If there is no single angle that describes **their relationship**, then they are not collinear.

There are several ways that two locations can be described as being non-collinear. One simple method is to say that they are opposite one another with respect to **some axis** or plane. For example, if one location is above the other on the same vertical plane, then they are not collinear. There are two locations, so they must be opposite one another.

.. An axis is a line on which points lie, especially if it is connected to **a geometric figure** such as a triangle. Because two points define a line, they are trivially collinear. A line can also be called parallel to itself; thus, the horizontal and vertical lines in any coordinate plane are examples of lines that are collinear.