Was trying to understand a hint of @Jean Marie to a previous question about orthocentric systems. It has to do with the fact that "the center of the incircle together with the three excircle centers form an orthocentric system". Therefore, it would be enough to prove it for such systems. Now we need a formula for these centers. I pulled one out of a classic text of complex numbers in plane geometry. It seems there was a problem with the sign, so I will let the readers decide if it is correct now. The statement is:
Let $a$, $b$, $c$ be complex numbers of equal absolute value. Then the center of the inscribed and exinscribed circles have the complex coordinates $$-(\sqrt{a} \sqrt{b} + \sqrt{a}\sqrt{c} + \sqrt{b}\sqrt{c})$$
Note that each complex square root has two values $\pm$. However the above expression has in fact $4$ values, as it should.
(Note that this implies
$$\sum_{k=0}^3 \vec{OI_k}=0$$ also $$\vec{EA}+\vec{EB}+\vec{EC}+\vec{EH}=0$$ where $O$ is the center of the circumscribed circle of a triangle $\Delta ABC$, $H$ is the orthocenter, and $E$ is the center of the nine-point circle. Known, but new to me).
I could not find it in other sources, but certainly it is known, so it is stated as a reference. Any feedback would be appreciated.
