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?

Let $z_1 / z_2=a+bi$ with $a,b \in \mathbb{R}$. Then $\bar z_1 / \bar z_2=a-bi$ and the given condition gives $(a+bi)+(a-bi) = 2 a = 2 \iff a = 1 \iff z_1/z_2 = 1 + bi$.
The angle $\angle ABO = \arg((z_1-z_2) / z_2)=\arg(z_1/z_2-1)=\arg(1+bi-1)=\arg(bi) = \pm \pi / 2$. Ignoring orientation, $\angle ABO = \pi / 2\,$. 
