When solving equations such as z^n+w=0, where w is a known complex number, i've noticed that whenever shown in an argand diagram the points representing z seem to form the vertices of a regular polygon. Does this always occur for any known integer n, where n>2?
I might have a go at proving the general case if I get bored with my revision. So far i've only shown that any two adjacent points representing z are of equal distance from each other, and that all points are equidistant from the origin. What else would I need to show for it to be a correct proof? In other words, what other properties are required for something to be a regular polygon? (other than the sides to be all the same length)