complex

i hav done this question, i further want to analyse it;

actually the question was if |Z1| = |Z2| = |z3| and z1 + z2 + z3 = 0 prove they are vertices of equilateral triangle:
i have done this part

i want knw if we can conclude that at least one of them is real or if they are the roots of x³= |z1|³

You cannot conclude whether they are real or not. But they will certainly form the vertices of an equilateral triangle

Let |z₁|=k.

Now, z_1+z_2+z_3=0

Also, \overline{z_1}+\overline{z_2}+\overline{z_3}=0 \Rightarrow k^2 \left(\frac{1}{z_1} + \frac{1}{z_2}+\frac{1}{z_3}\right)=0 \Rightarrow \frac{1}{z_1} + \frac{1}{z_2}+\frac{1}{z_3} = 0

So, z₁, z₂ and z₃ are roots of z^3-q = 0 where q is a complex number.

Let q = k e^{i \theta}

Then the three roots i.e. the three vertices are

k^{\frac{1}{3}} e^{i \frac{\theta}{3}}, \omega k^{\frac{1}{3}} e^{i \frac{\theta}{3}}, \omega^2 k^{\frac{1}{3}} e^{i \frac{\theta}{3}}

Its now obvious that the three vertices form an equilateral triangle. But none of them need be real.

there is one more way to say they forms vertices of equilateral triangle

its clear that z1,z2,z3 lie on a circle,the circumcenter of the triangle formed by z1,z2,z3 is the origin

further we have z1 + z2 + z3 = 0
so the centroid is also the origin

so it is equilateral triangle;)

@THEPROPHET: sir,what abt the hint given in the question i uploaded? how can we assume that one of them is real without any loss of generality?

WLOG You can assume z₁ is real, because multiplying each of the numbers by e^{i \theta} does not alter any of the conditions. Geometrically, you are rotating the figure by \theta. So you can rotate it appropriately so that one of the numbers become real. In fact you may assume it to be a positive real number.