Dear Alex,

By using property: midpoint of crossing segment of two circumcircles (O1) = XAB, (O2) = YBC is on circle (O) we can create a lot of concyclic points. Of course, we only choose points with interesting properties and interesting construction.
Concyclic point may be one alone (for example centroid U or intersection M). Some points are pair (for example N and P)
I suggest one more pair of concyclic points:
X* = intersection of XS with (O1) then X*, B, T' are collinear
Line NO intersects this collinear line X*T' at N' (O is center of our concyclic circle or midpoint of UB).
Easy to show that N' are on circle (O) and N' is reflection of N in center O.
Similarly we can construct P' = reflection of P in center O.
P' and N' also can be accepted as concyclic points.

I think we can find many more concyclic points. Some of them are very easy to prove, other are very hard.
The configuration with one segment, one point on the segment, two equilateral triangles can generate a lot of concyclic points are very similar with Archimedean Arbelos. In the arbelos we can construct a lot of Archimedean circles. In our configuration, we can construct a lot of concyclic points. Of course our configuration can not be interesting as arbelos.
As with arbelos, we can collect all our concyclic points in one catalogue, each point with its proof and its interesting properties. In this case, we must choose proper nomenclature (notations) for concyclic points such as P1, P2,... and may be name for them.
It is can be interesting play for learning how to discover geometry fact and then prove it.

What about generalized fact, I think it is very easy: we can always choose one parallel projection (in 3 dimension space) from one plane to another plane to get equilateral triangles from similar triangles. So what is true for equilateral triangles then it is true for general case and vice versa.

Best regards,
Bui Quang Tuan