Question

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

 

Answer 1

The condition for co linearity of three points${\displaystyle (x_1,y_1),(x_2,y_2)\text{and }(x_3,y_3)}$ is that the area enclosed by them should be equal to 0.

The formula for the area ‘A’ encompassed by three points ${\displaystyle (x_1,y_1),(x_2,y_2)\text{and }(x_3,y_3)}$   and  is given by the formula,

${\displaystyle A=\frac{1}{2}\left|\begin{array}{c}{x}_1-x_2 y_1-y_2\\ x_2-x_3 y_2-y_3\end{array}\right|}$

${\displaystyle A=\frac{1}{2}|(x_2-x_2)(y_2-y_3)-(x_2-x_3)(y_1-y_2)|}$

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

${\displaystyle \frac{1}{2}|(x_1-x_2)(y_2-y_3)-(x_2-x_3)(y_1-y_2)|=0}$

    ${\displaystyle |(x_1-x_2)(y_2-y_3)-(x_2-x_3)(y_1-y_2)|=0}$

The area ‘A’ encompassed by three points ${\displaystyle (x_1,y_1),(x_2,y_2)\text{and }(x_3,y_3)}$   ,  is also given by the formula,

${\displaystyle A=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|}$

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

${\displaystyle \frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|}$= 0

    ${\displaystyle x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)=0}$