In general, we can define the Hirst Inversion in Circumconic C, with pole Q as the map taking
the point M to the intersection of QM with the polar of M in C. In [ETC], the point Q is taken as
the perspector of C. In [CL32], it is taken as the centre of C. Here, we allow any choice of Q,
even on C. Clearly, each Hirst Inversion has order 2, and fixes the points of C. Algebraically,
the map takes M to a point M' whose coordinates are quadratic in those of M.
A Hirst Inversion is defined except at three points, the point Q and the contacts of C with the
tangents from Q. These are the "bad" points of the map. Any other point on the polar of Q will
map to Q. Any other point on one of the tangents maps to its contact point.
Theorem 1
The circumcubic K is self-inverse with respect to Hirst Inversion in circumconic C with pole Q
if and only if K is of the form pK(P<->Q,P), with P on C.
Proof
Suppose that K is a circumcubic. In general, the Hirst inverse K' will be a sextic. In order to get
a cubic image K', we must choose K containing three bad points. If Q is on C, we must take K
with C, K having a common tangent at Q.
Now suppose that K is self-inverse under this transformation. For M on K, since the line QM meets
the cubic K at most once more, and M' = M for M on C, we observe that QM will be tangent to K
at M if and only if M is on C. Thus we have
(1) QA, QB, QC are the tangents to K at A, B, C.
Then, as Q is on K, the cubic K is of the form pK(P<->Q,P).
(2) Now, as QP is the tangent at (the pivot) P, P is on C.
The final two tangents from Q are those at the other bad points.
The fact that these conditions are sufficient to make K self-inverse is easy by Maple.
Note
We have many different Hirst maps corresponding to the conics through P.
But each such map interchanges points on a line through Q, the secondary pivot.
It follows that all these maps have the same restriction to the cubic K, namely the
same as P-Ceva conjugation.
[CTC] Cubics in the Triangle Plane
[CL32] CL032 Hirst Pivotal Cubics
[E+G] Special Isocubics in the Triangle Plane
[ETC] Encyclopedia of Triangle Centers