[JEE Advanced 2005] A square circumscribes the circle $|z-1|=\sqrt2$. If one vertex is $2+\sqrt3\,i$, find the other vertices.

Answer: $(1-\sqrt3)+i,\quad -\sqrt3\,i,\quad (1+\sqrt3)-i$.

Vertices lie at distance $2$ from the centre $1$; $(2+\sqrt3i)-1=1+\sqrt3i=2e^{i\pi/3}$. Rotating by $\frac\pi2,\pi,\frac{3\pi}2$ about the centre gives $1-\sqrt3+i,\ -\sqrt3 i,\ 1+\sqrt3-i$.

JEE Advanced 2005 · Complex Numbers — verified solution by the Vidaara Team.