Question

# If three points $({x_1},{y_1}),({x_2},{y_2})$ and $({x_3},{y_3})$ lie on the same line then, prove that$\frac{{{y_2} - {y_3}}}{{{x_2}{x_3}}} + \frac{{{y_3} - {y_1}}}{{{x_3}{x_1}}} + \frac{{{y_1} - {y_2}}}{{{x_1}{x_2}}} = 0.$

According to question, three points $({x_1},{y_1}),({x_2},{y_2})$ and $({x_3},{y_3})$ lie on the same line. So we can say that the points are collinear. And we know that for three points to be collinear, following condition will hold:
${x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2}) = 0$
Dividing by ${x_1}{x_2}{x_3}$ on both sides of this equation, we’ll get:
$\Rightarrow \frac{{{x_1}({y_2} - {y_3})}}{{{x_1}{x_2}{x_3}}} + \frac{{{x_2}({y_3} - {y_1})}}{{{x_1}{x_2}{x_3}}} + \frac{{{x_3}({y_1} - {y_2})}}{{{x_1}{x_2}{x_3}}} = 0 \\ \Rightarrow \frac{{{y_2} - {y_3}}}{{{x_2}{x_3}}} + \frac{{{y_3} - {y_1}}}{{{x_3}{x_1}}} + \frac{{{y_1} - {y_2}}}{{{x_1}{x_2}}} = 0 \\ \$
$\Rightarrow \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \frac{{{y_3} - {y_2}}}{{{x_3} - {x_2}}}$
We will get the same condition as we have used earlier, ${x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2}) = 0$