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 [1].

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.

[1] Richard Deaux. Introduction to the Geometry of Complex Numbers.

-x only equals 0 if x equals 0 :)