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If z1,z2,z3 are three points lying on th...

If `z_1,z_2,z_3` are three points lying on the circle `|z|=2` then the minimum value of the expression `|z_1+z_2|^2+|z_2+z_3|^2+|z_3+z_1|^2=`

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The correct Answer is:
12
`|z|=|z_(2)| =|z_(3)| = 2`
Now, `|z_(1) + z_(2) + z_(3)|^(2) ge 0`
`rArr (z_(1) + z_(2) +z_(3))( barz_(1)+barz_(2) + barz_(3)) ge 0`
`rArr sum|z_(1)|^(2) +sum(z_(1)barz_(2) + barz_(1)z_(2)) ge 0`
`2sum|z_(1)|^(2)+sum(z_(1)barz_(2)+barz_(1)z_(2)) ge sum|z_(1)|^(2)`
`rArr |z_(1) +z_(2)|^(2)+|z_(2)+z_(3)|^(2)+|z_(3) + z_(1)|^(2) ge 12`
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