If $$z_1, z_2,z_3\in C$$, then $$z_1=z_2=z_3$$ is the __FILLWRAPl__ for the equation $$(z_1-z_2)^2+(z_2-z_3)^2=0$$ to hold true.

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If $$z_1, z_2,z_3\in C$$, then $$z_1=z_2=z_3$$ is the  __FILLWRAPl__ for the equation $$(z_1-z_2)^2+(z_2-z_3)^2=0$$ to hold true.

Uniteasy Admin · Accepted Answer

If $$z_1=z_2=z_3$$, then $$(z_1-z_2)^2+(z_2-z_3)^2=0$$. So it's true $$z_1=z_2=z_3$$ is the sufficient condition for the equation to hold true. However, if $$z_1 = i, z_2=0$$ and $$z_3=1$$, then 
$$(i-1)^2+(0-1)^2=i^2+1^2 = -1+1=0$$
which shows the equation is also equal to 0. So $$(z_1-z_2)^2+(z_2-z_3)^2=0$$ must not deduce the result of $$z_1=z_2=z_3$$. $$z_1=z_2=z_3$$ is not the necessary condition for the equation to hold true.

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