TITLE: Given four complex numbers $z_1, z_2, z_3$ and $z_4,$ show that they lie on a circle.
QUESTION [5 upvotes]: Given four complex numbers $z_1, z_2, z_3$, and $z_4,$ show that they lie on a circle if 
$$\arg\left(\frac{z_4-z_1}{z_4-z_2}\right)=\arg\left(\frac{z_3-z_1}{z_3-z_2}\right).$$
 How can I interpret this equality? And how can I show this statement? I need some help. Thanks.

REPLY [2 votes]: First of all,
$$\arg\left(\frac{z_4-z_1}{z_4-z_2}\right)$$
represents the angle $\angle Z_2 Z_4 Z_1$.  Similarly
$$\arg\left(\frac{z_3-z_1}{z_3-z_2}\right)$$
represents the angle $\angle Z_2 Z_3 Z_1$.  This is because to when you divide two complex numbers, you subtract the angles and we don't care about the magnitudes.  Euler's form for complex numbers makes it a bit clearer:
$$M_1 e^ {\theta_1} \div M_2 e^ {\theta_2} = \frac{M_1}{M_2}e^{\theta_1 - \theta_2}$$
Combine that with:
http://en.wikipedia.org/wiki/Inscribed_angle_theorem#Theorem
The $A$ and $B$ in the wikipedia graphic are your $z_1$ and $z_2$.