Is it possible to find a formula for $d$ in terms of $a$, $b$, and $c$?

Question

If $a$, $b$, $c$, and $d$ are complex numbers on the unit circle, and $\overline{ab}\perp\overline{cd}$, is it possible to find a formula for $d$ in terms of $a$, $b$, and $c$?

Answer 1

Hint: Chords $AB$ and $CD$ are perpendicular if and only if

$$ \stackrel{\frown}{AC} + \stackrel{\frown}{BD} = \stackrel{\frown}{CB} + \stackrel{\frown}{DA} $$

Hence, if you let $ a = e^{i \alpha}, b = e^{i \beta}, c = e^{i \gamma}, d = e^{i \delta}$, what can you say about $ \delta$ in terms of the rest?