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.

 Point $A: \quad$ $x_A$ = 0 , $y_A$ = 3 Point $B: \quad$ $x_B$ = 1.5 , $y_B$ = 0 Point $C: \quad$ $x_C$ = 3 , $y_C$ = -3 Decimal Places = 2

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

Slope
General Equation of a Line: ax + by = c .
Equations of Lines in Different Forms .
Online Geometry Calculators and Solvers .