Collinear Points
Collinear points are the group of three or more than three points that lie on the same straight line. It is not necessary that they are coplanar but they must lie on the same straight line. The word collinear is derived from the Latin words 'col' and 'linear' where col stands for together and linear means in the same line. The property of the points being collinear is known as collinearity. Let's learn more about collinear points along with a few solved examples.
1.  Introduction to Collinear Points 
2.  NonCollinear Points 
3.  Collinear Points Formula 
4.  Solved Examples on Collinear Points 
5.  Practice Questions 
6.  FAQs on Collinear Points 
Introduction to Collinear Points
Collinear points are the set of three or more points that exist on the same straight line. Collinear points may exist on different planes but not on different lines. The property of points being collinear is known as collinearity. So any three points or more will only be collinear if they are in the same straight line. Only one line is possible that can go through three different points which are collinear. In the image given below, points P, Q, and R are the collinear points.
NonCollinear Points
If three or more points do not lie on the same straight line, then they are said to be noncollinear points. If any point of all the points is not on the same line, then as a group they are noncollinear points. In the image given below, points M, N, O, P, and Q are noncollinear points since they do not lie on the same straight line.
Collinear Points Formula
The collinear points formula is used to find out whether three points are collinear or not. There are various methods that can tell whether three points are collinear or not. The three most common methods used to find if points are collinear or not are:
 Distance Formula
 Slope Formula
 Area of Triangle Formula
Let us discuss all of these formulas one by one:
Distance Formula
In this formula, we find the distance between the first and the second point, and then the distance between the second and the third point, and then check if the sum of these two distances is equal to the distance between the first and the third point. This will only be possible if the three points are collinear points. To calculate the distance between two points whose coordinates are known to us, we use the distance formula.
The distance between two points A(\( x_1, y_1\)) and B(\( x_2, y_2\)) is:
\(d=\sqrt{\left(x_{2}x_{1}\right)^{2}+\left(y_{2}y_{1}\right)^{2}}\)
So if we have three collinear points A, B, and C, then these points will be collinear if AB + BC = CA
Slope Formula
We apply the slope formula, to find the slope of lines formed by the 3 points under consideration. If the 3 slopes are equal, then the three points are collinear.
If we have three points X, Y, and Z, the points will be collinear if and only if: the slope of line XY = slope of line YZ = slope of line XZ. To calculate the slope of the line joining two points, we use the slope formula.
The slope of the line joining points P(\(x_1, y_1\)) and Q(\(x_2, y_2\)) is:
\(m = \dfrac{y_{2}y_{1}}{x_{2}x_{1}}\)
Area of Triangle Formula
In this method, we use the fact that a triangle formed by three collinear points will have no area since it will just be a line joining the three points. So if we have three collinear points, we assume that they form a triangle, and calculate the area of the triangle, and if we get the result as 0, then the points must be collinear points. So if three points are collinear, then the area of a triangle formed by the three points will be 0.
Area of the triangle with collinear points A(\(x_1, y_1\)), B(\(x_2, y_2\)), and C(\(x_3, y_3\)) as the vertices will be:
\( \text{A} =\frac{1}{2}\left\left(x_{1}\left(y_{2}y_{3}\right)+x_{2}\left(y_{3}y_{1}\right)+x_{3}\left(y_{1}y_{2}\right)\right)\right=0\)
Related Topics
Check out these interesting articles to learn more about collinearity and its related topics
 X and Y Graph
 Coordinate Plane
 Equation of a Straight Line
 Area of the triangle in coordinate geometry
 Distance between Two Points
 Equation of a Line
Important Notes
 Three points will be collinear, if only if they fall in the same straight line.
 This property of points being collinear is known as collinearity.
 Collinear points can exist on different planes.
Solved Examples on Collinear Points

Example 1:
By using the slope formula, find out whether the points P(1, 2), Q(2, 3), and R(3, 4) are collinear or not.
Solution:
To check, we are using the slope formula and find the slope of any two pairs of lines.
Let us join find the slope of the lines RQ and QP, and check if we get the slopes equal to each other. If they are equal, then the points will be collinear.
Slope of line RQ is
\(\begin{align*} m_{1} &=\dfrac{y_{3}y_{2}}{x_{3}x_{2}} \\ m_{1} &=\dfrac{43}{32} \\ m_{1} &=\dfrac{1}{1} \\ m_{1} &=1 \end{align*}\)
Slope of line QP is
\(\begin{align*} m_{2} &=\dfrac{y_{2}y_{1}}{x_{2}x_{1}} \\ m_{2} &=\dfrac{23}{12} \\ m_{2} &=\dfrac{1}{1} \\ m_{2} &=1 \\ m_{1} &=m_{2} \end{align*}\)
As the slope of both lines are equal, the points are collinear.
Answer: P(1, 2), Q(2, 3), and R(3, 4) are collinear points.

Example 2:
Check whether the given points are collinear or not: A (−3, −1), B (−1, 0), and C (1, 1).
Solution:
Given:
(\(x_1, y_1\)) is (3, −1)
(\(x_2, y_2\)) is (1, 0)
(\(x_3, y_3\)) is (1, 1)If the points are collinear, then:
Area of triangle =\(\frac{1}{2} \mid\left(x_{1}\left(y_{2}y_{3}\right)+x_{2}\left(y_{3}y_{1}\right)+x_{3}\left(y_{1}y_{2}\right) \mid=0\right.\)On substituting the values, we get:
\(\dfrac{1}{2} \mid(3(01)+ (1)(1(1))+1(10) \mid=0 \\ \dfrac{1}{2}(3 2 1) =0 \\ 0=0 \)
As the area of the triangle is 0, the points are collinear.
Answer: The points A(−3, −1), B (−1, 0), and C (1, 1) are collinear.
FAQs on Collinear Points
What are Collinear Points in Triangle?
According to Euler's theory, in a triangle, there exists an Euler line, in which three points of concurrency of the triangle lies. The three points which lie on the Euler's line are the orthocenter, the circumcenter, the centroid. Hence the given points of concurrencies of the triangle are the collinear points in a triangle.
How Do You Know if Points are Collinear?
To know whether the points are collinear or not we use various formulas. The basic and most commonly used formulas are
 Distance Formula
 Slope Formula
 Area of Triangle Formula
with the help of these formulas, we find out whether the points are collinear or not.
 In the distance formula, we find the distance between the first and the second point, and then the distance between the second and the third point, and then check if the sum of these two distances is equal to the distance between the first and the third point. This will only be possible if the three points are collinear points.
 We apply the slope formula, to find the slope of lines formed by the 3 points under consideration. If the 3 slopes are equal, then the three points are collinear.
 And in the area of the triangle formula, we use the fact that a triangle formed by three collinear points will have no area since it will just be a line joining the three points. So if we have three collinear points, we assume that they form a triangle, and calculate the area of the triangle, and if we get the result as 0, then the points must be collinear points.
What Does Collinear Mean?
The word collinear is derived from the Latin words 'col' and 'linear' where col stands for together and linear means in the same line. Collinear points are the group of three or more than three points that lie on the same straight line. It is not necessary that they are coplanar but they must lie on the same straight line.
Are 2 Points Collinear?
Yes, two points are always collinear since you can draw a straight line between any two points. There exist no two such points that a straight line cannot pass through them, therefore any two points are always collinear points.
What are NonCollinear Points?
If three or more points do not lie on the same straight line, then they are said to be noncollinear points. If any point of all the points is not on the same line, then as a group they are noncollinear points. For noncollinear points, the area of the triangle joined by the three points will always be greater than 0.