Let a, b, and c be three complex numbers.
These numbers form the vertices of an equilateral triangle in the complex plane if and only if
This theorem can be found in .
If we rotate the matrix above, we multiply its sign by -1. If we then swap two rows we multiply the determinant again by -1. So we could write the criterion above with the 1’s on the top row.
See also this post which gives the area of a triangle in the complex plane, also in terms of a determinant.
 Richard Deaux. Introduction to the Geometry of Complex Numbers.