All isogonal circum-strophoids with a node at I are nK with their root R on the trilinear polar of the Gergonne point X(7) containing the points X(241), X(514), X(650), X(665), X(905), X(1323), X(1375), X(1465), X(1638), X(3002), X(3004), X(3008), X(3015). See Special Isocubics §4.3.2. For non-isogonal circum-strophoids, see CL038. See also CL001, CL027 and a generalization in the page P-conical cubics.
Each cubic is the locus of foci of conics centered on a line L passing through the incenter I = X(1) and another point P = (p : q : r).
It is also spK(Z, X1) where Z is the infinite point of L, see CL055.
The singular focus F lies on the circumcircle of ABC since it is the isogonal conjugate of Z.
These strophoids form a pencil of circular cubics whose equation is :
(a^2 y z + b^2 z x + c^2 x y) ∑ p (c y - b z) = (x + y + z) ∑ b c p x (c y - b z),
where ∑ p (c y - b z) = 0 is the equation of the line passing through I and P,
and ∑ b c p x (c y - b z) = 0 is the equation of the circumconic which is the isogonal conjugate of the previous line i.e. the circumconic passing through I and P*.
The following table shows a selection of such remarkable strophoids.
R(515) = (b - c)( -a^2 so + b^2 sc + c^2 sb + abc) : :
R(758) = a(b - c)(2 SA + bc) : :