# Collinear Points Free Online Calculator

More that two points are ** collinear ** if they are on the same line.

Given three points \( A \), \( B \) and \( C \), an online calculator to calculate the slopes of the line through \( A \) and \( B \), and the line through \( B \) and \( C \) and hence decide whether the three points are collinear or not.

## Formulas Used in Calculator

The three points \( A(x_A,y_A) \), \( B(x_B,y_B)\) and \( C(x_C,y_C) \) are collinear if the slopes of the lines through any two points are equal.The slope \( m_{AB} \) of line through \( A \) and \( B \) is given by

\( m_{AB} = \dfrac{y_B - y_A}{x_B-x_A} \)

The slope \( m_{AC} \) of line through \( A \) and \( C \) is given by

\( m_{AC} = \dfrac{y_C - y_A}{x_C-x_A} \)

The slope \( m_{BC} \) of line through \( B \) and \( C \) is given by

\( m_{BC} = \dfrac{y_C - y_B}{x_C-x_B} \)

The equation of the line through the points \( A \) and \( B \) may be written as

\( y - y_B = m_{AB}(x - x_B ) \)

For point \( C \) to be on the line through the points \( A \) and \( B \), the following equation may be satisfied

\( y_C - y_B = m_{AB}(x_C - x_B ) \)

which may be written as

\( m_{AB} = \dfrac{y_C - y_B}{x_C - x_B} = m_{BC}\)

Conclusion: For the three points to be collinear, we need to satify the following

\( m_{AB} = m_{BC} \)

Example

Are the points \( A(-1,5) \) , \( B(1,1) \) and \( C(3,-3) \) collinear?

Solution

The slope \( m_{AB} \) of line through \( A \) and \( B \) is given by

\( m_{AB} = \dfrac{y_B - y_A}{x_B-x_A} = \dfrac{1 - 5}{1-(-1)} = - 2 \)

The slope \( m_{BC} \) of line through \( B \) and \( C \) is given by

\( m_{BC} = \dfrac{y_C - y_B}{x_C-x_B} =\dfrac{-3 - 1}{3-1} = -2 \)

Hence \( m_{BC} = m_{AB} \) and therefore the three points are collinear

## Use of Online Calculator to Verify that three Given Points Are Collinear

Enter the coordintes of the three points \( A \),\( B\) and \( C \) as real numbers and press "Calculate".

The results are: the slopes \( m_{AB} \) and \( m_{BC} \) and the conclusion whether the three points are collinear or not.

## Results

## Activities

Use the calculator to find slopes \( m_{AB} \), \( m_{BC} \) and verify that the three points are collinear. Then calculate the slopes \( m_{AB} \), \( m_{BC} \) and \( m_{AC} \) analytcally and verify that they are all equal.

a)
\( A(-5,-2) \),
\( B(-2,1) \)
,
\( C(2,5) \).

b)
\( A(-5,7) \),
\( B(-1,-1) \)
,
\( C(1,-5) \).

c)
\( A(0,3) \),
\( B(2,2) \)
,
\( C(6,0) \).

## More References and Links

SlopeGeneral Equation of a Line: ax + by = c .

Equations of Lines in Different Forms .

Online Geometry Calculators and Solvers .