Strange numbers

Algebra Level 5

Let $a$ , $b$ , and $c$ be non-zero real numbers such that

$(ab+bc+ca)^3 = abc (a+b+c)^3.$

Which of the following is definitely true about $a$ , $b$ and $c$ ?

Relevant wikis:

Arithmetic sequence.
Geometric sequence.
Harmonic sequence.

They are in a geometric sequence They are in a harmonic sequence They are in two or more of the other choices None of the other choices They are in a arithmetic sequence