Question

If $|z_1|=|z_2|=|z_3|$ and $arg\left(z_2/z_1\right)={60}^{\circ }$, find the degree measure of $arg\left(\frac{z_2-z_3}{z_1-z_3}\right)$.

Answer 1

Since, $|z_1|=|z_2|=|z_3|$
Therefore, we can say that $z_1$, $z_2$ & $z_3$ lie on a circumcircle with center origin.
Since, $arg\left(z_2/z_1\right)={60}^{\circ }$
Therefore, $arg\left(z_3/z_1\right)=arg\left(z_3/z_2\right)={60}^{\circ }$
Using the property of circumcircle $arg\left(\frac{z_3-z_1}{z_2-z_1}\right)=\frac{1}{2}arg\left(\frac{z_3}{z_2}\right)={30}^{\circ }$