How does this expression in complex variable hold?

Question

If $z_1, z_2, z_3$ are vertices of an equilateral triangle then $$|z_1-z_2|=|z_2-z_3|=|z_3-z_1|$$. But how does $\frac{1}{z_1-z_2}+\frac{1}{z_2-z_3}+\frac{1}{z_3-z_1}=0$ hold?

Answer 1

We have $$\frac{1}{z_i-z_j}=\frac{\overline{z_i-z_j}}{|z_i-z_j|^2}\quad \text{and}\quad \overline{z_i-z_j}=\overline{z_i}-\overline{z_j}.$$ With the use of $|z_1-z_2|=|z_2-z_3|=|z_3-z_1|$ we get $$\frac{1}{z_1-z_2}+\frac{1}{z_2-z_3}+\frac{1}{z_3-z_1}=\frac{1}{|z_1-z_2|^2}\left({\overline{z_1}-\overline{z_2}+\overline{z_2}-\overline{z_3}+\overline{z_3}-\overline{z_1}}\right)=0$$