Question

If $\overrightarrow{PO}+\overrightarrow{OQ}=\overrightarrow{QO}+\overrightarrow{OR}$, prove that the points $P,Q,R$ are collinear.

Answer 1

$\overrightarrow{PO}+\overrightarrow{OQ}=\overrightarrow{QO}+\overrightarrow{OR}$

$\Rightarrow \overrightarrow{O}-\overrightarrow{P}+\overrightarrow{q}-\overrightarrow{O}=\overrightarrow{O}-\overrightarrow{q}+\overrightarrow{r}-\overrightarrow{O}$

$\Rightarrow \overrightarrow{q}-\overrightarrow{p}=\overrightarrow{r}-\overrightarrow{q}$

$\Rightarrow 2\overrightarrow{q}=\overrightarrow{p}+\overrightarrow{r}$

$\Rightarrow \overrightarrow{q}=\frac{\overrightarrow{p}+\overrightarrow{r}}{2}$

$\Rightarrow Q$ is mid-point of $PR$

$\Rightarrow P.Q,R$ are collinear.