Question

If three points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ lie on the same line, prove that
$\frac{y_2-y_3}{x_2{x}_3}+\frac{y_3-y_1}{x_3{x}_1}+\frac{y_1-y_2}{x_1{x}_2}=0$

Answer 1

Points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are collinear

$(x_1{y}_2+x_2{y}_3+x_3{y}_1)-(x_2{y}_1+x_3{y}_2+x_1{y}_1)=0$

$\Rightarrow (x_1{y}_2-x_1{y}_3)+(x_2{y}_3-x_2{y}_1)+(x_3{y}_1-x_3{y}_2)=0$

$\Rightarrow x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)=0$

$\Rightarrow \frac{x_1(y_2-y_3)}{x_1{x}_2{x}_3}+\frac{x_2(y_3-y_1)}{x_1{x}_2{x}_3}+\frac{x_3(y_1-y_2)}{x_1{x}_2{x}_3}=0$

$\Rightarrow \frac{(y_2-y_3)}{x_2{x}_3}+\frac{(y_3-y_1)}{{x_{1}x}_3}+\frac{(y_1-y_2)}{x_1{x}_2}=0$