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

Explanation

Let O be the centre of a regular octagon ABCDEFGH.
Then `vec(OA)` = `-vec(OE)`, `vec(OB)` = `-vec(OF)`
`vec(OC)` = `-vec(OG)`, `vec(OD)` = `-vec(OH)`
Now `vec(OA)+vec(OB)+vec(OC)+vec(OD)+vec(OE)+vec(OF)+vec(OG)+vec(OH)`
=`(vec(OA)+vec(OE)) + (vec(OB)+vec(OF)) + (vec(OC)+vec(OG)) + (vec(OD)+vec(OH))`
= `vec0+vec0+vec0+vec0` = 0