Pronunciation: /koʊˈlɪn i ər/ ?

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Manipulative 1: Any two points are collinear. Click on the blue points and drag them to change the figure. The two blue points can be placed anywhere in figure and still define exactly one line.

If points are collinear, they are in the same line1. Since two points define a line, any two points are collinear (see figure 1). If three points are collinear, a line drawn using any two of the points will contain the third (see figure 2).

If objects are non-collinear, no one line can be drawn that contains all the objects (see figure 3).

Three points that are on the same line.
Figure 1: Collinear points

Three points that are not on the same line.
Figure 2: Non-collinear points

Related Words

Collinearity: Having to do with whether or not objects are in the same line.

How to tell if Points are Collinear

x-y graph with the points (-1,-1), (1,0), and (3,1) marked.
Figure 3: Graph with 3 points.

In 2-dimensional analytical geometry, each point has an x-coordinate and a y-coordinate. These coordinates can be used to see if three are collinear. To see if three or more points are collinear, pick one of the points as a reference point. If the slopes of the lines defined by the reference point and each of the other points are equal, the points are collinear. Table 1 shows the steps for determining if the points in figure 3 are collinear.

1m = (y1-y0)/(x1-x0)This algorithm starts with the equation of a line in point slope form.
2(-1,-1), (3,1)Start by selecting any two of the points. For this demonstration, select (-1,-1) and (3,1).
3m=(y2-y1)/(x2-x1)=(1-(-1))/(3-(-1))=2/4=1/2Use the slope formula to determine the slope of the line defined by the two points.
4(1,0), (3,1)Now pick one of the points already used and the third point.
5m=(y2-y1)/(x2-x1)=(1-0)/(3-1)=1/2Find the slope of the line defined by those two points.
61/2 = 1/2Since the slopes are equal, the three points are collinear.
Table 1: How to find out if three points are collinear.


  1. collinear. WordNet. Princeton University. (Accessed: 2011-01-08).
  2. Casey, John, LL.D., F.R.S.. The First Six Books of the Elements of Euclid, pg 5. Casey, John, LL.D. F.R.S.. Hodges, Figgis & Co., 1890. (Accessed: 2010-01-02).
  3. Palmer, Claude Erwin and Taylor, Daniel Pomeroy. Solid Geometry, pg 280. Scott, Foresman and Company, 1918. (Accessed: 2010-01-12).

Cite this article as:

Collinear. 2010-01-12. All Math Words Encyclopedia. Life is a Story Problem LLC.


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Revision History

2010-03-15: Improved instructions for manipulative 3. Added section on analytical geometry. (McAdams, David.)
2010-01-12: Added "References" (McAdams, David.)
2008-07-05: Added More Information (McAdams, David.)
2007-08-23: Initial version (McAdams, David.)

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