Question

If $S$ is the circumcenter, $O$ is the orthocenter of $△ABC$, then $\overrightarrow{SA}+\overrightarrow{SB}+\overrightarrow{SC}=$

• A. $2\overrightarrow{OS}$
• B. $2\overrightarrow{SO}$
• C. $\overrightarrow{OS}$
• D. $\overrightarrow{SO}$

Answer 1

The correct option is D $\overrightarrow{SO}$

A very important property to be remembered of a triangle is that the centroid divides the line joining orthocentre and circumcentre in the ratio $2:1$
$\Rightarrow C=\frac{2S+O}{3}$

$\Rightarrow \frac{\overline{A}+\overline{B}+\overline{C}}{3}=\frac{2\overline{S}+\overline{O}}{3}$

$\Rightarrow \overline{A}+\overline{B}+\overline{C}=2\overline{S}+\overline{O}$

Thus, $\overline{SA}+\overline{SB}+\overline{SC}=\overline{A}+\overline{B}+\overline{C}-3\overline{S}=\overline{SO}$