Prove that the sum of all vectors drawn from the centre of a regular octagon to its vertices is the zero vector.

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Solution

Given: A regular octagon of eight sides with centre O.
To show: $\vec{O A} + \vec{O B} + \vec{O C} + \vec{O D} + \vec{O E} + \vec{O F} + \vec{O G} + \vec{O H} = \vec{0} .$
Proof: We know centre of the regular octagon bisects all the diagonals passing through it.
$\vec{O A} = - \vec{O E}, \vec{O B} = - \vec{O F}, \vec{O D} = - \vec{O H}$ and $\vec{O C} = - \vec{O G} .$
$\Rightarrow \vec{O A} + \vec{O E} = \vec{0}, \vec{O B} + \vec{O F} = \vec{0}, \vec{O D} + \vec{O H} = \vec{0}$ and $\vec{O C} + \vec{O G} = \vec{0} .$ ............(i)
Now,
$\vec{O A} + \vec{O B} + \vec{O C} + \vec{O D} + \vec{O E} + \vec{O F} + \vec{O G} + \vec{O H} = (\vec{O A} + \vec{O E}) + (\vec{O B} + \vec{O F}) + (\vec{O C} + \vec{O G}) + (\vec{O D} + \vec{O H}) = \vec{0} + \vec{0} + \vec{0} + \vec{0} = \vec{0}$
Hence proved.

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