lean_workbook_plus_59309

A shorter approach. Since $|z_k| = 1$ , it follows $\dfrac {1} {z_k} = \overline{z_k}$ . Thus the relation $\dfrac {z_1} {z_2} + \dfrac {z_2} {z_3} + \dfrac {z_3} {z_1} = 1$ writes as $z_1\overline{z_2} + z_2\overline{z_3} + z_3\overline{z_1} = 1$ . By taking the conjugate we then also have $\overline{z_1}z_2 + \overline{z_2}z_3 + \overline{z_3}z_1 = 1$ , or $1 = \dfrac {z_2} {z_1} + \dfrac {z_3} {z_2} + \dfrac {z_1} {z_3} = \dfrac {z_1} {z_2} \cdot \dfrac {z_2} {z_3} + \dfrac {z_2} {z_3} \cdot \dfrac {z_3} {z_1} + \dfrac {z_3} {z_1} \cdot \dfrac {z_1} {z_2}$ . Finally, $\dfrac {z_1} {z_2} \cdot \dfrac {z_2} {z_3} \cdot \dfrac {z_3} {z_1} = 1$ .