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	?

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