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Let $\mathrm{ABC}$ be an acute scalene triangle, and $\mathrm{O}$ and $\mathrm{H}$ be its circumcentre and orthocenter respectively. Further let $\mathrm{N}$ be the midpoint of OH. The value of the vector sum $\overrightarrow{\mathrm{NA}}+\overrightarrow{\mathrm{NB}}+\overrightarrow{\mathrm{NC}}$ is
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The correct answer is:
$\frac{1}{2} \overrightarrow{\mathrm{HO}}$
Circumcenter (origin $\mathrm{O}$ )
$(\overrightarrow{\mathrm{O}})=\left(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{3}\right) ; \mathrm{H}=\left(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{2}\right)$
$\overrightarrow{\mathrm{N}}=\frac{1}{4}(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}})$
$(\overrightarrow{\mathrm{O}})=\left(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{3}\right) ; \mathrm{H}=\left(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{2}\right)$
$\overrightarrow{\mathrm{N}}=\frac{1}{4}(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}})$
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