# Can you show that if (x_1+x_2)/x_3=(y_1+y_2)/y_3, then (x_1,y_1), (x_2,y_2), and (x_3,y_3) are linear?

## Can you show that if $\frac{{x}_{1} + {x}_{2}}{x} _ 3 = \frac{{y}_{1} + {y}_{2}}{y} _ 3$, then $\left({x}_{1} , {y}_{1}\right)$, $\left({x}_{2} , {y}_{2}\right)$, and $\left({x}_{3} , {y}_{3}\right)$ are linear?

Dec 19, 2015

They are not collinear for all ${P}_{1} , {P}_{2} , {P}_{3}$.

#### Explanation:

Let ${P}_{i} = \left({x}_{i} , {y}_{i}\right) \mathmr{and} r$ be the line ${P}_{1} {P}_{2}$.

$r : \left(x , y\right) = {P}_{1} + t \left({P}_{2} - {P}_{1}\right)$

${y}_{3} = {x}_{3} \frac{{y}_{1} + {y}_{2}}{{x}_{1} + {x}_{2}}$

P_3 in r Leftrightarrow exists t in RR ; (x_3, y_3) = (x_1, y_1) + t(x_2-x_1, y_2-y_1)

${x}_{3} = {x}_{1} + t \left({x}_{2} - {x}_{1}\right) \mathmr{and} {x}_{3} \frac{{y}_{1} + {y}_{2}}{{x}_{1} + {x}_{2}} = {y}_{1} + t \left({y}_{2} - {y}_{1}\right)$

$t = \frac{{x}_{3} - {x}_{1}}{{x}_{2} - {x}_{1}} \mathmr{and}$

${x}_{3} \frac{{y}_{1} + {y}_{2}}{{x}_{1} + {x}_{2}} = {y}_{1} + \frac{{x}_{3} - {x}_{1}}{{x}_{2} - {x}_{1}} \left({y}_{2} - {y}_{1}\right)$

$R i g h t a r r o w {x}_{3} \left({y}_{1} + {y}_{2}\right) \left({x}_{2} - {x}_{1}\right) = {y}_{1} \left({x}_{1} + {x}_{2}\right) \left({x}_{2} - {x}_{1}\right) + \left({x}_{1} + {x}_{2}\right) \left({x}_{3} - {x}_{1}\right) \left({y}_{2} - {y}_{1}\right)$

$R i g h t a r r o w {x}_{2} {x}_{3} {y}_{1} + {x}_{2} {x}_{3} {y}_{2} - {x}_{1} {x}_{3} {y}_{1} - {x}_{1} {x}_{3} {y}_{2} = {x}_{2}^{2} {y}_{1} - {x}_{1}^{2} {y}_{1} + \left({x}_{1} {x}_{3} - {x}_{1}^{2} + {x}_{2} {x}_{3} - {x}_{2} {x}_{1}\right) {y}_{2} + \left(- {x}_{1} {x}_{3} + {x}_{1}^{2} - {x}_{2} {x}_{3} + {x}_{2} {x}_{1}\right) {y}_{1}$

$R i g h t a r r o w 0 = \left(2 {x}_{1} {x}_{3} - {x}_{1}^{2} - {x}_{2} {x}_{1}\right) {y}_{2} + \left(- 2 {x}_{2} {x}_{3} + {x}_{2} {x}_{1} + {x}_{2}^{2}\right) {y}_{1}$