TITLE: The roots of the equation $z^n=(1+z)^n$...
QUESTION [5 upvotes]: The complex roots of the equation 

$$z^{n}=(1+z)^{n}$$

$A.$ are vertices of a regular polygon
$B.$ lie on a circle
$C.$ are collinear
$D.$ none of these
Don't know where to start.. Please help.!
NOTE:  $z$ is a complex number.

REPLY [7 votes]: An easier method, stolen from comment to an answer from a similar thread.
If $z$ satisfies $z^n=(1+z)^n$, then $|z|^n=|1+z|^n$ and: $$|z|=|z+1|$$
The geometric interpretation is that the distance (in the complex plane) from $0$ to $z$ equals the distance from $-1$ to $z$. The locus of points with the same distance from $0$ and $-1$ is the vertical line: $$\mathrm{Re}\ z = -\frac12$$ so every solution to the original equation lies on this line.