Test whether three complex numbers are vertices of an equilateral triangle

Let a, b, and c be three complex numbers.

Triangle with vertices at exp(pi/12), exp(9pi/12), and exp(17pi/12).

These numbers form the vertices of an equilateral triangle in the complex plane if and only if

\left| \begin{matrix} a & b & 1 \\ b & c & 1 \\ c & a & 1 \\ \end{matrix} \right| = 0

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.

\left| \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ b & c & a \\ \end{matrix} \right| = 0

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.

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