Given three points in the complex plane (i.e. numbers $z\_1,z\_2,z\_3\in\mathbb C$), they define a unique circle (unless they are collinear). When does that circle have radius one?

I know how to compute that “the hard way”, i.e. by separating real and imaginary part. From there I could either construct perpendicular bisectors and intersect these, or I could solve $ax\_k+by\_k+c=x\_k^2+y\_k^2$ for $k\in\{1,2,3\}$ and then deduce the radius from the $a,b,c$ I found.

But I guess there might be some more elegant way to express this condition using vocabulary more suited for complex numbers. Separating numbers into real and imaginary part for all numbers should not be needed, even though conjugation might still be needed at some point.

As a motivating example: we [know](https://math.stackexchange.com/a/599456/35416) that four points are cocircular iff they satisfy

$$\begin{vmatrix}
x\_1^2+y\_1^2 & x\_1 & y\_1 & 1 \\
x\_2^2+y\_2^2 & x\_2 & y\_2 & 1 \\
x\_3^2+y\_3^2 & x\_3 & y\_3 & 1 \\
x\_4^2+y\_4^2 & x\_4 & y\_4 & 1
\end{vmatrix}=0$$

but with $z\_k=x\_k+iy\_k$ you can also check the condition

$$\frac{(z\_1-z\_3)(z\_2-z\_4)}{(z\_1-z\_4)(z\_2-z\_3)}\in\mathbb R$$

which is a lot easier to write and compute. I'm looking for some similar simplification for the case of unit radius.