# Geometric significance of the terms of the relation $a^2+b^2+c^2=ab+bc+ca$ that holds when complex $a$, $b$, $c$ form an equilateral triangle?

It is known (and can be easily shown) that if complex numbers $$a, b, c$$ form an equilateral triangle on the complex plane, then $$a^2+b^2+c^2=ab+bc+ca$$

Question Is there a geometric significance/interpretation of

• the squares of each of these numbers, as well their sum (i.e. LHS), and
• the product of pairs of each of the numbers, as well as their sum (i.e. RHS)?
• Hi @Hypergeometricx, you didn't wish us a new year this time but this is a nice question here :) I'm also curious for the geometric interpretation of given expression apart from $\sum (a-b)^2 = 0$ – cosmo5 Feb 23 at 14:11
• See whether it helps. – Ng Chung Tak Feb 23 at 14:12
• – Albus Dumbledore Feb 23 at 14:15
• Hi @cosmo5 - thanks for noticing the previous new year puzzles haha. Hope you liked them. Busy with other stuff but might come up with some in the future, you never know. :) – Hypergeometricx Feb 24 at 2:09
• Thanks for your comments and references so far, everyone. The question is specifically about the geometric significance of the terms as shown, rather than that of other rearranged forms. – Hypergeometricx Feb 24 at 2:10