Question

If three points $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ 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

$(x_1,y_1),(x_2,y_2),(x_3,y_3)$ lie on same line

To prove

$\frac{{}_2-y_3}{x_2{x}_3}+\frac{y_3-y_1}{x_3{x}_1}+\frac{y_1-y_2}{x_1{x}_2}=0$

Let the line be $y=mx+c$

$(x_1,y_1)$ is on the line

$\Rightarrow y_1=m{x}_1+c$

$(x_2,y_2)$ is on the line

$\Rightarrow y_2=m{x}_2+c$

$(x_3,y_3)$ is on the line

$\Rightarrow y_3=m{x}_3+c$

LHS

$\Rightarrow \frac{y_2-y_3}{x_2{x}_3}+\frac{y_3-y_1}{x_3{x}_1}+\frac{y_1-y_2}{x_1{x}_2}$

$\Rightarrow \frac{(m{x}_2+c-m{x}_3-c)}{x_2{x}_3}+\frac{(m{x}_3+c-m{x}_1-c)}{x_1{x}_3}+\frac{(m{x}_1+c-m{x}_2-c)}{\left(x_1{x}_2\right)}$

$\Rightarrow m[\frac{(x_2-x_3)}{x_2{x}_3}+\frac{(x_3-x_1)}{x_3{x}_2}+\frac{(x_1-x_2)}{x_1{x}_2}]$

$\Rightarrow m[\frac{(x_1{x}_2-x_1{x}_3)}{x_1{x}_2{x}_3}+\frac{(x_2{x}_3-x_2{x}_1)}{x_1{x}_2{x}_3}+\frac{(x_1{x}_3-x_2{x}_3)}{x_1{x}_2{x}_3}]$

$\Rightarrow m\left(\frac{0}{x_1{x}_2{x}_3}\right)\Rightarrow 0\Rightarrow RHS$

$\Rightarrow LHS=RHS$