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

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)]
\(\mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}=2 \mathbf{O}^{\prime} \mathbf{O}\)