Test whether three complex numbers are vertices of an equilateral triangle

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.

The post Test whether three complex numbers are vertices of an equilateral triangle first appeared on John D. Cook.