How to approach this complex numbers question? I have started to approach this problem from writing the point on the perpendicular bisector as (z2+z3)/2 + ix(z2-z3), then was thinking of equaling the distance or something to find x. But I am not able to proceed, I am new to complex number, and don't know the real tricks how to approach the question. What some basic theorems/forumales are to be used here?


Since the circum circle is $|z|=1$, the orthocenter is given by $z_1+z_2+z_3$. If $z_4$ represents the point $P$, then 
\begin{align*}
\frac{z_1+z_2+z_3 - z_1}{\overline{z_1+z_2+z_3} - \bar{z_1}} &= \frac{z_4-z_1}{\bar{z_4}-\bar{z_1}}\\
z_2z_3 &= -z_4z_1
\end{align*}
where we have used $\bar{z} = \frac{1}{z}$ when $|z|=1$ in the last step. Thus 
$z_4 = -\frac{z_2z_3}{z_1}$. Since $D$ is the mid point of $HP$, it follows that $D$ is given by 
\begin{align*}
\frac{1}{2}\left(z_1+z_2+z_3 -\frac{z_2z_3}{z_1}\right)
\end{align*}
