Conditions To Make Complex Numbers $z_1, z_2, z_3, z_4$ Vertices of a Square

Question

Let $z_1,z_2,z_3,z_4\in\mathbb C$ be distinct. State conditions in terms of computation of complex numbers, which make $z_1,z_2,z_3,z_4$ vertices of a square (in the counterclockwise direction).

---

Since $z_1,z_2,z_3,z_4$ are going to be vertices of a square, I know I need the distance between $z_1$ and $z_2$ equal to the distance between $z_2$ and $z_3$, and so on... Would a correct way to state this be: \begin{equation} |z_1-z_2|=|z_2-z_3|=|z_3-z_4|=|z_4-z_1| \end{equation}

Another condition I can think of is that the distance between $z_1$ and $z_3$ be equal to the distance of $z_2$ and $z_4$, namely \begin{equation} |z_1-z_3|=|z_2-z_4|=\sqrt{2|z_1-z_2|^2} \end{equation}

Those are the only two conditions I can think of. Are they written correctly, reasonable, and am I missing anything?

Thanks!

Answer 1

So that you don't have to square or take square roots.

Notice that $i(z_2-z_1)=z_3-z_2$ and $i(z_3-z_2)=z_4-z_3$ is enough. Geometrically $a-b$ is a translation of the vector that goes from $b$ to $a$. Geometrically, multiplying by $i$ is a $\pi/2$ rotation counterclockwise.