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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)?
Hypergeometricx
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    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 '21 at 14:11
  • See whether it helps. – Ng Chung Tak Feb 23 '21 at 14:12
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    Also see https://math.stackexchange.com/questions/1068254/let-z-1-z-2-and-z-3-be-complex-vertices-of-an-equilateral-triangle-show – Albus Dumbledore Feb 23 '21 at 14:15
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    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 '21 at 02:09
  • 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 '21 at 02:10
  • Rewrite the equation as sum of squares and than you get three vertices of triangle - three vectors at circular form. Is this a geometrical interpretation that satisfies you? – Moti Feb 26 '21 at 04:57

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