How to describe the locus of $z$ where $\arg \left(\frac{z-z_1}{z-z_2}\right)=\theta$. I need to describe the locus of $z$ which stisfies $\displaystyle\arg \left(\frac{z-z_1}{z-z_2}\right)=\theta$. 


$\displaystyle\arg \left(\frac{z-z_1}{z-z_2}\right)=\theta \Rightarrow \arg(z-z_1)-\arg(z-z_2)=\theta$

So I got that  ,

$\arg(z-z_1)=\arg(z-z_2)+\theta$

So $z$ lies on an arc. 

If  $0&lt;\theta \leq\frac{\pi}{2}$ , $z$ lies on the major arc of a circle which passes through $z_1$ and $z_2.$ But there are two such circles ! How can we  identify the correct circle ? 

If  $\frac{\pi}{2}&lt;\theta &lt; \pi$ , $z$ lies on the major arc of a circle which passes through $z_1$ and $z_2.$ But there are two such circles ! How can we  identify the correct circle ?
Since $\theta$ is a given real number, the arc is oriented. Only one of two possible circles has the right orientation of the angle from $z_1$ via $z$ to $z_2.$
Since $\theta$ is a given real number, the arc is oriented. Only one of two possible circles has the right orientation of the angle from $z_1$ via $z$ to $z_2.$