# Write the condition of collinearity of points

Question:

Write the condition of collinearity of points (x1, y1), (x2, y2) and (x3, y3).

Solution:

The condition for co linearity of three points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ is that the area enclosed by them should be equal to 0 .

The formula for the area 'A' encompassed by three points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{1}, y_{3}\right)$ is given by the formula,

$\mathrm{A}=\frac{1}{2}\left|\begin{array}{ll}x_{1}-x_{2} & y_{1}-y_{2} \\ x_{2}-x_{3} & y_{2}-y_{3}\end{array}\right|$

$\mathrm{A}=\frac{1}{2}\left|\left(x_{1}-x_{2}\right)\left(y_{2}-y_{3}\right)-\left(x_{2}-x_{3}\right)\left(y_{1}-y_{2}\right)\right|$

Thus for the three points to be collinear we need to have,

$\frac{1}{2}\left|\left(x_{1}-x_{2}\right)\left(y_{2}-y_{3}\right)-\left(x_{2}-x_{3}\right)\left(y_{1}-y_{2}\right)\right|=0$

$\left|\left(x_{1}-x_{2}\right)\left(y_{2}-y_{3}\right)-\left(x_{2}-x_{3}\right)\left(y_{1}-y_{2}\right)\right|=0$

The area ' $A$ ' encompassed by three points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ is also given by the formula,

$\mathrm{A}=\frac{1}{2}\left|x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right|$

Thus for the three points to be collinear we can also have,

$\frac{1}{2}\left|x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right|=0$

$x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$