The simplest figure in geometry is referred to as a point. It does not have a certain size and is often represent with a dot and labeled using capital letters. Points are in all geometric figures and we define space to be the set of all points. As such single points have no characteristics, when the points are grouped, we get different types of points coplanar, non-coplanar, collinear and non-collinear.  

Points lying on one line are called as Collinear. Any two points are ever collinear because when connected, they determine a line.

Three or more points can be collinear but it is not necessary that all the points has to be collinear.
In the above figure P, Q, and R are collinear points.

Points Picture

In Cartesian plane any three points are collinear (say the points are A($x_1, y_1$) B($x_2, y_2$) and C($x_3, y_3$) if the ratios $\frac{(y_2 - y_1)}{(x_2 - x_1)}$ = $\frac{(y_3 – y_1)}{(x_3 – x_1)}$ = $\frac{(y_3 – y_2)}{(x_3 – x_2)}$ are equal.

In other words the slope of line formed by any two given points is same.

Example: Determine if the points A(2, 6), B(-1, 0) and C(5, 12) are collinear.
Solution:Let us determine slope formed by pair of points say A and B.

A(2, 6) B(-1, 0)
slope of line joining AB = $\frac{(0 - 6)}{(-1 - 2)}$

= $\frac{-6}{-3}$

Slope = 2

Check slope of line joining B and C.
B(-1, 0) and C(5, 12)
Slope of line joining B and C = $\frac{(12 - 0)}{(5-(-1)}$

= $\frac{12}{6}$

Slope = 2

Since slope of lines joining AB and BC are equal, the points A, B and C are collinear.


Non-collinear points


These are the points which will not lie on the same plane, in the above figure X, Y, and Z are example for non-collinear points.

If three points are given say X Y and Z and slopes of lines formed by the pairs of points do not match then the points are not lying on the same line. Such points are called as non collinear.

Example: Determine if the points X(-2, -4), y(0, -1) and Z(3, 5)
Solution: To check if the three points are collinear let us determine the sope of lines formed by pair of points say X-Y and Y-Z.
Slope of line joining X and Y
X(-2, -4) and Y(0, -1)

slope = $\frac{(-1 - (-4))}{(0 - (-2))}$

= $\frac{3}{2}$

Determine the slope of line joining Y and Z

Y(0, -1) and Z(3, 5)

Slope = $\frac{(5 - (-1))}{(3 - 0)}$

= $\frac{6}{3}$ = 2

Since the slope of XY is not equal to the slope of YZ the points are not lying on the same line and hence are non collinear.
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