Let ABC be an acute scalene triangle,and O an | vedclass

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(C) Let the position vectors of the vertices of $	riangle ABC$ be $\vec{a}, \vec{b},$ and $\vec{c}$ respectively.Let the circumcentre $O$ of $	riang

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$\frac{1}{2} \overrightarrow{HO}$

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(C) Let the position vectors of the vertices of $	riangle ABC$ be $\vec{a}, \vec{b},$ and $\vec{c}$ respectively.Let the circumcentre $O$ of $	riangle ABC$ be the origin,so $\vec{O} = \vec{0}$.The position vector of the orthocentre $H$ is given by $\vec{H} = \vec{a} + \vec{b} + \vec{c}$.Since $N$ is the mid-point of $OH$,its position vector $\vec{n}$ is given by $\vec{n} = \frac{\vec{O} + \vec{H}}{2} = \frac{\vec{a} + \vec{b} + \vec{c}}{2}$.We need to find the sum $\overrightarrow{NA} + \overrightarrow{NB} + \overrightarrow{NC}$.This is equal to $(\vec{a} - \vec{n}) + (\vec{b} - \vec{n}) + (\vec{c} - \vec{n})$.$= (\vec{a} + \vec{b} + \vec{c}) - 3\vec{n}$.Substituting $\vec{n} = \frac{\vec{a} + \vec{b} + \vec{c}}{2}$,we get:$= (\vec{a} + \vec{b} + \vec{c}) - 3 \left( \frac{\vec{a} + \vec{b} + \vec{c}}{2} ight)$.$= (\vec{a} + \vec{b} + \vec{c}) \left( 1 - \frac{3}{2} ight) = -\frac{1}{2} (\vec{a} + \vec{b} + \vec{c})$.Since $\vec{H} = \vec{a} + \vec{b} + \vec{c}$,this is equal to $-\frac{1}{2} \vec{H} = -\frac{1}{2} \overrightarrow{OH} = \frac{1}{2} \overrightarrow{HO}$.

Let $ABC$ be an acute scalene triangle,and $O$ and $H$ be its circumcentre and orthocentre respectively. Further,let $N$ be the mid-point of $OH$. The value of the vector sum $\overrightarrow{NA}+\overrightarrow{NB}+\overrightarrow{NC}$ is

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