Pairs of Inverse Cubics

A table giving pairs of cubics inverse with respect to the Circumcircle.

For each row the cubics are pK(W1,P1) and pK(W2,P2).
Each pivot Px is on the Euler Line. Each pole Wx is on the Brocard Axis.
Hence each Wx is a barycentric product X3*Ex, with Ex on the Euler Line.
Note that, as X3 is on each cubic, Ex is its isoconjugate, so on the cubic.
Thus the cubic meets the Euler Line (only) in X3, Px, Ex.

In each row, P1,P2, and E1,E2 are inverse with respect to the Circumcircle.
W1,W2 are inverse with respect to the circumconic with perspector X184.

Given Px ≠ X30 on the Euler Line, we can construct Wx as follows:
Let L be the perpendicular bisector of PxX110.
Then Wx is the isogonal conjugate of the isotomic conjugate of the tripole of L.
Alternatively,
Wx is the barycentric product of Px and the inverse in the Circumcircle of the
isogonal conjugate of Px. Thus this inverse is the secondary pivot.

Given Wx ≠ X6 on the Brocard Axis, we can construct Px as follows:
Let L be the tripolar of the isotomic conjugate of the isogonal conjugate of Wx.
Then Px is the reflection of X110 in L.

 number W1 P1 E1 W2 P2 E2 notes 1 X50 X3 X186 X6 X30 X4 K073,K001 2 X32 X23 X25 X187 X2 X468 K108,K043 3 X571 X186 X24 X3003 X4 X403 ??, K339 4 X577 X2071 X3 X3*X30 X20 X30 5 X3 X858 X2 X3*X23 X22 X23 6 X3053 X468 X459 X3*? X25 ?