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Question
If \[\overrightarrow{AO} + \overrightarrow{OB} = \overrightarrow{BO} + \overrightarrow{OC} ,\] prove that A, B, C are collinear points.
Solution
We have,
\[\overrightarrow{AO} + \overrightarrow{OB} = \overrightarrow{BO} + \overrightarrow{OC} . \]
\[ \Rightarrow \overrightarrow{AO} - \overrightarrow{BO} = \overrightarrow{OC} - \overrightarrow{OB} . \]
\[ \Rightarrow \overrightarrow{OB} - \overrightarrow{OA} = \overrightarrow{OC} - \overrightarrow{OB} . \]
\[ \Rightarrow \overrightarrow{AB} = \overrightarrow{BC} .\]
Hence A, B and C are collinear points.
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