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