Given: a regular octagon

To prove the sum of all vectors drawn from the centre of a regular octagon to its vertices is the zero vector

Proof:

Let O be the centre of a regular octagon, we know that the centre of a regular octagon bisects all the diagonals passing through it as shown in figure below

Thus,

The sum of all vectors drawn from the centre of a regular octagon to its vertices is

Substitute the values from eqn(i) in above eqn, we get

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

Hence, proved.