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