Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$.
Find a simple expression for $d$ in terms of $a$, $b$, and $c$.
I have thought of using power of a point, but that has gotten me nowhere so far. Can anyone help me?
Best Answer
We want that $$\lambda:={d-c\over b-a}$$ is purely imaginary. This is equivalent with $\lambda=-\bar\lambda$, or $${d-c\over b-a}=-{\bar d-\bar c\over\bar b-\bar a }=-{{1\over d}-{1\over c}\over{1\over b}-{1\over a}}\ .$$ This at once simplifies to $ab=-cd$.