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If in a $\Delta A B C, \mathbf{O}$ and $\mathbf{O}^{\prime}$ are the incentre and orthocentre respectively, then $\left(\boldsymbol{O}^{\prime} \mathbf{A}+\boldsymbol{O}^{\prime} \mathbf{B}\right.$
$\left.+\mathbf{O}^{\prime} \mathbf{C}\right)$ is equal to
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$\left.+\mathbf{O}^{\prime} \mathbf{C}\right)$ is equal to
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Verified Answer
The correct answer is:
$2 \mathbf{O}^{\prime} \mathbf{O}$
O' $\mathbf{A}=\mathbf{O}^{\prime} \mathbf{O}+\mathbf{O A}$
$\mathbf{O}^{\prime} \mathbf{B}=\mathbf{O}^{\prime} \mathbf{O}+\mathbf{O B}$
$\mathbf{O}^{\prime} \mathbf{C}=\mathbf{O}^{\prime} \mathbf{O}+\mathbf{O C}$
$\Rightarrow \mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}$
$=3 \mathbf{0} \mathbf{0}+(\mathbf{O} \mathbf{A}+\mathbf{O B}+\mathbf{O C}) \ldots(\mathrm{i})$
$\because \quad \mathbf{O A}+\mathbf{O B}+\mathbf{O C}=\mathbf{O O}^{\prime}=-\mathbf{O}^{\prime} \mathbf{O}$
$\therefore \quad \mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}=3 \mathbf{O}^{\prime} \mathbf{O}-\mathbf{O}^{\prime} \mathbf{O}$
[from Eq. (i)]
$\mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}=2 \mathbf{O}^{\prime} \mathbf{O}$
$\mathbf{O}^{\prime} \mathbf{B}=\mathbf{O}^{\prime} \mathbf{O}+\mathbf{O B}$
$\mathbf{O}^{\prime} \mathbf{C}=\mathbf{O}^{\prime} \mathbf{O}+\mathbf{O C}$
$\Rightarrow \mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}$
$=3 \mathbf{0} \mathbf{0}+(\mathbf{O} \mathbf{A}+\mathbf{O B}+\mathbf{O C}) \ldots(\mathrm{i})$
$\because \quad \mathbf{O A}+\mathbf{O B}+\mathbf{O C}=\mathbf{O O}^{\prime}=-\mathbf{O}^{\prime} \mathbf{O}$
$\therefore \quad \mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}=3 \mathbf{O}^{\prime} \mathbf{O}-\mathbf{O}^{\prime} \mathbf{O}$
[from Eq. (i)]
$\mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}=2 \mathbf{O}^{\prime} \mathbf{O}$
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