Given three complex numbers $|Z_1|= 2 , |Z_2|= 3, |Z_3| = 4$ find the maximum value of $$|Z_1-Z_2|^2 +|Z_2-Z_3|^2+|Z_3-Z_1|^2$$

If we treat them as three vectors $a, b, c$ centred at zero the above expression becomes $$2(|a|^2+|b|^2+|c|^2)-2(a\cdot b + b\cdot c + c\cdot a)$$
I've been unsuccessful trying to find the minimum value of $a\cdot b + b\cdot c + c\cdot a$.


marked as duplicate by Winther, Abcd, callculus, Lee David Chung Lin, blub Apr 15 at 19:40

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    $\begingroup$ There probably do exist some more elegant ways to do it with some clever use of inequalities, but in general this can be solved using Lagrange multipliers. $\endgroup$ – lisyarus Apr 15 at 13:08
  • $\begingroup$ You're correct with your approach, more on that, it can be written as $3(|a|² + |b|² + |c|²) - (a + b + c)²$ and since there vectors are always linearly independent in a plane, the second term is zero. I had posted it in chat, don't know if you saw. $\endgroup$ – Ice Inkberry Apr 19 at 16:33

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