If $A(z_1)$ and $(z_2)$ are two points in argand plane, find $\angle ABO$ If $A(z_1)$ and $(z_2)$ are two points in argand(complex) plane such that $$\frac{z_1}{z_2}+\frac{\overline{z_1}}{\overline{z_2}}=2$$. Find the value of $\angle ABO$ where $O$ is origin.
Using given condition, I found that Real part of $\frac{z_1}{z_2}=1$ but I am not able to use this to find $\angle ABO$. Could someone help me with this?

Assume the points $O,A,B$ are distinct.

Let $A = z_1 = s + ti$ and $B = z_2 = u + vi$. Then 
$$\frac{z_1}{z_2} + \frac{\overline{z_1}}{\overline{z_2}} =2 $$
$$\Rightarrow \frac{s + ti}{u + vi} + \frac{s - ti}{u - vi} = 2$$
Combining and simplifying the above equation yields
$$su + tv = u^2 + v^2$$
$$\Rightarrow u^2 + v^2 - su - tv = 0$$
$$\Rightarrow (u - s/2)^2 + (v - t/2)^2 = (s^2 + t^2)/4$$
$$\Rightarrow |z_2 - z_1/2| = |z_1|/2$$
Then $0,z_1,z_2$ are equidistant from $z_1/2$,
hence, since $O,A,B$ are distinct, it follows that $B$ is on the semicircle with diameter $OA$. Therefore angle $ABO$ is a right angle.
