Let $A,B, C$ the vertices of a triangle in the complex plane. Why I can supposse that the afixes of $A, B, C$ can be $u^2$, $v^2$ and $w^2$ s.t. $|u|=|v|=|w|=1$ and $-uv-vw-uw=-i$?
I saw this ideea in a proof of an exercise and I can't understand why.
Depending on the problem, sometimes the lengths of the sides are not important, only the ratios. Then the first equation just lets you know that you can choose the origin of the complex plane to be in the circumcenter of the triangle. Then, in principle, you can choose $A=u=e^{i\phi_A}$ and similar to $B$ and $C$. The second equation is just a phase of the triangle. You can rotate the triangle by an angle $\phi$, so $u'=ue^{i\phi}$ (similar for $v$ and $w$). Then the left side of your second equation is just multiplied by $e^{2i\phi}$.