Prove that the sum of the vectors from the centre to the vertices of a
regular hexagon is 0
Let's call the centre $O$ and the vertices are $A, B, C, D, E$ and $F$. Therefore, the sum in the question is $OA + OB + OC + OD + OE + OF$. How do I prove with basic vectors formulae that this sum is equal to zero? I know you can probably prove it using symmetry etc. but is it possible to prove it my way? I tried modifying the equation in various ways, e.g. ended up with $6OA + 5AB + 4BC + 3CD + 2DE + EF$ but it does not seem to help.
Best Answer
Suppose the sum were something other than zero. In that case, you could rotate your setup by $60°$ around the center of the hexagon. As a result the sum should rotate along, but since rotating the hexagon by $60°$ doesn't change it, you end up in the original situation. The only vector which doesn't change under a $60°$ rotation is the zero vector.
If you don't thave that $60°$ rotational symmetry established, you can pick any other symmetry you may have proved before. If you take the point reflection in the center (i.e. a $180°$ rotation), this becomes very similar to the pairing of opposite vectors some comments suggest.