If $|z_{1}|=2,|z_{2}|=3,|z_{3}|=4$,then find maximum value of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$.
My attempt:I considered 3 circles having centre origin and radii as $2,3,4$. Then I tried invoking the triangle inequality $|z_{1}-z_{2}|\leq|z_{1}|+|z_{2}|$ where equality occurs if origin,$z_{1},z_{2}$ are collinear with origin lying between $z_{1}$ and $z_{2}$. This means that $z_{1}$ and $z_{2}$ lie lie on a common diameter with origin lying between $z_{1}$ and $z_{2}$ and thus maximum value of $|z_{1}-z_{2}|$ will be $2+3=5$. If I apply the same reasoning to $|z_{2}-z_{3}|$, I get $3+4=7$.
So now I have $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$.
$\leq 5^2+7^2+|z_{3}-z_{1}|^2$.
Not able to proceed from here bcoz if I apply the same logic it will put $z_{1}$ and $z_{3}$ on different sides of origin due to which $z_{1}$ and $z_{2}$ will lie on same side of origin wich is in contradiction to the very first assumption that origin,$z_{1},z_{2}$ are collinear with origin lying between $z_{1}$ and $z_{2}$.
