Dear Alex,

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...

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

Best regards,
Bui Quang Tuan

>Dear Alex,
>
>We can construct similar closed chain of six concyclic
>points as following:
>
>A1A2A3 is a triangle and O is circumcenter.
>L1 = line OA1
>L2 = line OA2
>L3 = line OA3
>
>We start with any point P1 on the line L1
>Perpendicular from P1 to A1A2 cuts L2 at P2
>Perpendicular from P2 to A2A3 cuts L3 at P3
>Perpendicular from P3 to A3A1 cuts L1 at P4
>...
>At the end P7=P1 and six points P1, P2, P3, P4, P5, P6 are
>on one circle centered at O.
>
>If instead of circumcenter O we take orthocenter H then
>P7=P1 but six points are not concyclic.
>
>I think the fact P7=P1 is true if instead of O we take any
>point on Darboux cubic. This cubic contains: incenter,
>circumcenter, orthocenter... But this fact need a lot of
>calculation.
>
>Only in circumcenter case six points are concyclic.
>
>In the incenter case: P1, P2, P3, P4, P5, P6 are centers of
>touching circles. They are not concyclic but we can
>construct touching circles and touching points are
>concyclic.
>
>Best regards,
>Bui Quang Tuan