>I have calculated and the locus of points such that we can
>make closed chain of six points is union of circumcircle and
>Darboux cubic which contains: incenter, circumcenter,
>orthocenter...

Have you tried drawing the lines which are not perpendicular but fall under the same inclination?

>Two cases circumcenter, orthocenter are very trivial so we
>have here only interesting case with incenter.

Yes, I think I can handle the circumcenter and the orthocenter. These come naturally.