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My question is to make a sketch of the complex plane showing a typical pair of complex numbers $z_1$ and $z_2$ which satisfy these equations: $$z_2-z_1 = (z_1-a)e^{2i\frac{\pi}3}$$ $$a-z_2 = (z_2-z_1)e^{2i\frac{\pi}3}$$ where $a$ is a real positive constant. Then also to describe the geometrical figure whose vertices are $z_1,$ $z_2$, and $a$.

I'm not really sure where to start with this. Thank you for any help.

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Firstly, have a look at the figure below.

Explanation:

First, take the module on both sides of the two equations:

$ |z_2-z_1| = |z_1-a| \ $ and $ \ |a - z_2| = |z_2-z_1| \ $

Thus $a,z_1,z_2$ are at the same distance one from the others: they constitute an equilateral triangle.

Moreover, multiplying the first equation by $e^{i \pi/3}$ gives:

$$(z_2-z_1)e^{i\frac{\pi}3} = (z_1-a)\underbrace{e^{3i\frac{\pi}3}}_{-1} \ \ $$

which is equivalent to:

$$(z_1-a) = (z_1-z_2)e^{i\frac{\pi}{3}} $$

Geometrical interpretation of this relationship : $\vec{AC}$ is obtained from $\vec{AB}$ by a $+\pi/3$ rotation, giving the only possible figure.

enter image description here

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