How to Find if the Points are Collinear?

There are three basic ways of finding if three points are collinear or not. There are various ways to estimate whether the three points are parallel or not but we will discuss the three most often used formulae to determine whether three points are collinear or not. The following formulas for collinear points are-

• Distance Formula Method
• Slope Formula Method
• Area of Triangle Formula Method

Distance Formula Method

We apply the distance formula to determine the difference between three points whose coordinates are known and lie on the same line. This is the very least used method to calculate the collinearity of given points.

Let’s consider the points A, B and C with their coordinates (x₁, y₁), (x₂, y₂) and (x₃, y₃) respectively are three collinear points, then,

Distance from A to B + Distance from B to C = Distance from A to C

AB + BC = AC

√{(x₂ -x₁ )² + (y₂ -y₁ )² + √ (x₃ -x₂ )² + (y₃ – y₂ )²} = √{(x₃ -x₁ )²+(y₃ -y₁)²}

Hence, A, B and C points are collinear.

Learn more about Distance Formula.

Slope Formula Method

Slope method is used to determine whether all three points are definitely on the same line or not by using their coordinates. In simple words, these three points are collinear if their all slopes are equal. Slope method are most accurate method to determine the collinear points. Three points A, B, and C, will only be collinear if 

Slope of line AB = Slope of line BC = Slope of line CA.

Let’s consider the points A, B and C with their coordinates (x₁, y₁), (x₂, y₂) and (x₃, y₃) respectively lying on the same line.

Hence, slope of line AB = slope of line BC= slope of line CA

m_AB = m_BC = m_CA

OR

(y₂ -y₁ )/(x₂ -x₁ ) = (y₃ -y₂ )/(x₃ -x₂ ) = (y₃ -y₁ )/(x₃ -x₁ )

Here, m denotes the slope of the line.

Read more about Slope of the Line.

Area of Triangle Method

Area of Triangle method is used to determine the collinearity of three points if their area is equal to zero. In simple words, the triangle created by three points whose coordinates are known will only be collinear if it does not contain any area.

Let’s consider the points A, B and C of a triangle with their coordinates (x₁, y₁), (x₂, y₂) and (x₃, y₃) respectively :

Area of Triangle (⧍ABC) = 0

OR

1/2[x₁(y₂ – y₃ ) + x₂ (y₃ – y₁ ) + x₃ (y₁ – y₂ ) = 0

Also, Check

• Equation of Line in 3D
• Parallel Lines
• Perpendicular Lines

Collinear Points are sets of three or more than three points that lie in a straight line. In simple words, if three or more points are collinear, they can be connected with a straight line without any change in slope.

In this article, we will discuss the concept of collinear points, collinear point definition, collinear point meaning, and properties. We will also know how to determine the three points collinearity by different methods. Further, we will also solve various examples and provide practice questions for a better understanding of the concept of this article.