Find all complex numbers $z$ such that $|z|=\frac{1}{|z|}=|1-z|$ 
Problem: Find all complex numbers $z$ such that $|z|=\frac{1}{|z|}=|1-z|$.

Basically I have an idea how to solve this and I get $x=\frac12$ but how should I express it mathematically? Should I go and find $y$ also?

Hint: $\require{cancel}$ $|z|^2=|1-z|^2$ $\iff$ $z \bar z = (1-z)(1-\bar z)$ $\iff$ $\cancel{z \bar z}=1-z-\bar z+\cancel{z \bar z}\,$
Since $1=|z|^2=z \bar z$ it follows that $\bar z = \cfrac{1}{z}$ so the last equality becomes $z^2-z+1=0\,$, which is a simple quadratic having the two possible values of $z\,$ as roots.
