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Collinear Points
Collinear points are those points that lie on the same straight line. It is not necessary that they should be 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 us learn more about collinear points in this article.
1.  What are Collinear Points? 
2.  NonCollinear Points 
3.  Collinear Points Formula 
4.  FAQs on Collinear Points 
What are Collinear Points?
Collinear points are a 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. Observe the figure given below in which 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 figure 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 determine whether three points are collinear or not. The three most common formulas that are used to find if points are collinear or not are the Slope Formula, the Area of Triangle Formula, and the Distance Formula. Let us discuss all these formulas one by one.
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.
For example, if we have three points X, Y, and Z, the points will be collinear 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 cannot be formed by three collinear points. This means if any 3 points are collinear they cannot form a triangle. Therefore, we check the points of the triangle by using them in the formula for the area of a triangle. If the area is equal to 0, then those points will be considered to be collinear. In other words, the triangle formed by three collinear points will have no area since it will just be a line joining the three points. The formula for the area of a triangle that is used to check the collinearity of points is expressed as:
Area of the triangle with the given points (vertices) A(x_{1}, y_{1}), B(x_{2}, y_{2}), and C(x_{3}, y_{3}) is:
\( \text{A} =\frac{1}{2}\left\left(x_{1}\left(y_{2}y_{3}\right)+x_{2}\left(y_{3}y_{1}\right)+x_{3}\left(y_{1}y_{2}\right)\right)\right=0\)
Distance Formula
Using 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. After this, we 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 in the order A, B, and C, then these points will be collinear if AB + BC = CA.
All the three methods can be understood with the help of the solved examples given under the Practice Section.
Tips on Collinear Points:
 Three points will be collinear, 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.
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Collinear Points Examples

Example 1:
By using the slope formula, find out whether the points P(1, 2), Q(2, 3), and R(3, 4) are collinear points or not.
Solution:
To check the collinearity of points, we will use the slope formula and find the slope of any two pairs of lines formed by the points
Let us find the slope of the lines PQ and QR, and check if we get the slopes equal to each other. If they are equal, then the points will be collinear.
Slope of line PQ is
\(\begin{align*} m_{2} &=\dfrac{y_{2}y_{1}}{x_{2}x_{1}} \\ m_{2} &=\dfrac{32}{21} \\ m_{2} &=\dfrac{1}{1} \\ m_{2} &=1 \\ m_{1} &=m_{2} \end{align*}\)
Slope of line QR is
\(\begin{align*} m_{1} &=\dfrac{y_{3}y_{2}}{x_{3}x_{2}} \\ m_{1} &=\dfrac{43}{32} \\ m_{1} &=\dfrac{1}{1} \\ m_{1} &=1 \end{align*}\)
As the slope of both the lines is 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:
A(x_{1}, y_{1}) is (3, −1)
B(x_{2}, y_{2}) is (1, 0)
C(x_{3}, y_{3}) is (1, 1)Let us use the area of triangle formula to check the collinearity of the given points. If the points are collinear, then the area of the triangle will be 0:
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.

Example 3: Use the distance formula to check whether the given points are collinear points or not: P(8, 11), Q(2, 3), R(1, 1).
Solution: If the distance of PQ + distance of QR = distance of PR, then the three points P, Q, and R will be collinear. So, let us find the distance between the points P and Q using the distance formula where x_{1} = 8, y_{1} = 11, x_{2} = 2, y_{2} = 3
PQ = √[(x_{2}  x_{1})^{2} + (y_{2 } y_{1})^{2}]
PQ = √[(2  8)^{2} + (3_{ } 11)^{2}]
PQ = √[( 6)^{2} + (8)^{2}]
PQ = √(36 +^{ }64) = √100 = 10
Now, let us find the distance between Q and R using the distance formula where, x_{1} = 2, y_{1} = 3, x_{2} = 1, y_{2} = 1
QR = √[(x_{2}  x_{1})^{2} + (y_{2 } y_{1})^{2}]
QR = √[(1  2)^{2} + (1_{ } 3)^{2}]
QR = √[(3)^{2} +^{ }(4)^{2})] = √25 = 5
PQ + QR = 10 + 5 = 15 units
Now, let us find the distance between P and R using the distance formula where, x_{1} = 8, y_{1} = 11, x_{2} = 1, y_{2} = 1
PR = √[(1  8)^{2} + (1_{ } 11)^{2}]
PR = √[(9)^{2} + (12)^{2}]
PR = √(81 +^{ }144) = √225 = 15 units
Therefore, it can be seen that PQ + QR = 10 + 5 = 15 units, and the distance between P and R = 15 units. This means that the given points are collinear.
FAQs on Collinear Points
What are Collinear Points in Geometry?
Collinear points are a 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.
How to Find Collinear Points?
There are various methods that are used to find out whether three points are collinear or not. The three most common methods used to find out the collinearity of points is by using the distance formula, the slope formula, and the area of triangle formula. Using these formulas, we find out whether the points are collinear or not. A detailed explanation is given under the heading 'Collinear Point Formula' on this page.
 Distance Formula: Using 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. Then, we 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.
 Slope Formula: We apply the slope formula to find the slope of the lines formed by the 3 points under consideration. If the 3 slopes are equal, then the three points are collinear.
 Area of Triangle Formula: Using the area of the triangle formula, we apply 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. Therefore, we check the points of the triangle by using them in the formula for the area of a triangle. If the area is equal to 0, then those points will be considered to be collinear.
How to Name Collinear Points?
Collinear points are named in the usual way that is used to label any point using capital letters, however, they are mentioned together as a group. This means if points A, B, and C form a straight line then we say that points A,B, and C are collinear.
How to Prove that 3 Points are Collinear?
There are many ways in which any 3 points can be proved to be collinear. One of the methods is by using the formula for the area of a triangle. We substitute the coordinates of all the 3 points in this formula. If the area comes to 0, this proves that the three points are collinear. The formula that is used is,
\( \text{A} =\frac{1}{2}\left\left(x_{1}\left(y_{2}y_{3}\right)+x_{2}\left(y_{3}y_{1}\right)+x_{3}\left(y_{1}y_{2}\right)\right)\right=0\)
where the given points A(x_{1}, y_{1}), B(x_{2}, y_{2}), and C(x_{3}, y_{3}) are the vertices of the triangle.
What are the Collinear Points in a Triangle?
According to the Euler's theory, in a triangle, there exists a straight line called the Euler's line, which passes through the orthocenter, the circumcenter, and the centroid of the triangle. Hence, these given points of concurrencies of the triangle are the collinear points in a triangle.
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 Two Points Always Collinear?
Yes, two points are always collinear since we can draw a straight line between any two points. There exist no two such points through which a straight line cannot pass. 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 the given 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.
What is the Difference Between Collinear and NonCollinear Points?
Collinear points are two or more points that lie on a straight line whereas noncollinear points are points that do not lie on one straight line.
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