Question

If $\overline{a},\overline{b},\overline{c}$ are position vectors of vertices $A,B,C$ of $\Delta ABC$. If $\overline{r}$ is position vector of a point $P$ such that $(|\overline{b}-\overline{c}|+|\overline{c}-\overline{a}|+|\overline{a}-\overline{b}|)\overline{r}$ $=|\overline{b}-\overline{c}|\overline{a}+|\overline{c}-\overline{a}|\overline{b}+|\overline{a}-\overline{b}|\overline{c}$ then the point $P$ is

• A. centroid of $\Delta ABC$
• B. Orthocentre of $\Delta ABC$
• C. circumcentre of $\Delta ABC$
• D. incentre of $\Delta ABC$

Answer 1

The correct option is D incentre of $\Delta ABC$

The above equation is the definition of an incentre of a triangle .
The point of intersection of angle bisectors of a triangle gives the incentre.