[ND]: The point with barycentrics
[(s-a)*(|a-b|-|a-c|)² : (s-b)*(|b-c|-|b-a|)² :
and search number 7.11040574869
I think is not in ETC and
as Paul mentioned lies on the incircle.
*** I think if a < b < c, this is the point
((s-a)(b-c)^2 : (s-b)(c+a-2b)^2 : (s-c)(a-b)^2)
which does not lie on the incircle.
Could be characterized as X(11) and X(1317)
a 3-degree polynomial center ?
I think we can find several such points.