Complex numbers with Maxima

I got a better solution for this one $|z_1-z_2|^{2}$ + $|z_3-z_2|^{2}$ + $|z_3-z_1|^{2}$ $= 2(|z_1|^{2}+|z_2|^{2}+|z_3|^{2}) - (\overline{z_1}.z_2+\overline{z_2}.z_1+\overline{z_3}.z_2 + \overline{z_2}.z_3 + \overline{z_3}.z_1 + \overline{z_1}.z_3) = 3(|z_1|^{2}+|z_2|^{2}+|z_3|^{2}) - |z_1+z_2+z_3|^{2}$ so this means we need to choose 3 complex numbers such that $z_1+z_2+z_3 = 0$ so this gives us the maximum value $3(|z_1|^{2}+|z_2|^{2}+|z_3|^{2})$ = $\boxed{87}$

Just an outline of my solution: I consider the three triangles with their vertices at the origin and at $z_1,z_2,z_3$ . Applying the law of cosines, we see that the sum we seek to maximize is $2*2^2+2*3^2+2*4^2-12\cos x-16\cos y-24\cos z$ where $x+y+z=2\pi$ . Using Lagrange multipliers, we find that the minimum of $12\cos x+16\cos y+24\cos z$ subject to the constraint $x+y+z=2\pi$ is $-29$ . Thus the maximum is $8+18+32+29=\boxed{87}$