Prove that the solutions to $(z+1)^7 = z^7$ have same real parts Prove that all the solutions to the equation $(z+1)^7=z^7$ have equal real parts, and find what this real part is equal to.

I found that $z=\frac{1}{e^{2\pi i/7}-1}, \frac{1}{e^{4\pi i/7}-1}, \frac{1}{e^{6\pi i/7}-1}, \frac{1}{e^{8\pi i/7}-1}, \frac{1}{e^{10\pi i/7}-1}, \frac{1}{e^{12\pi i/7}-1},$ but I'm not sure what to do next.

Based on the proposed equation, one concludes that
\begin{align*}
(z + 1)^{7} = z^{7} & \Rightarrow |z + 1| = |z|\\\\
& \Rightarrow |z + 1|^{2} = |z|^{2}\\\\
& \Rightarrow z\overline{z} + z + \overline{z} + 1 = z\overline{z}\\\\
& \Rightarrow 2\operatorname{Re}(z) = -1\\\\
& \Rightarrow \operatorname{Re}(z) = -\frac{1}{2}
\end{align*}
and we are done.
Hopefully this helps !
