The general case of Hirst inversion is considered briefly in Hirst Inversion and Cubics.

The case where the pole is the centre of the conic is considered in Hirst Inversion - Isocubics.

Here, we look at the case where the pole is the perspector of the circumconic. This is the version

which appears in ETC - Glossary. We look at cubics of type pK(W,P) and nK0(W,P) since the two

theories have much in common.

**Notation**

C(Q) is the circumconic with perspector Q.

T(X) is the tripolar of X.

X&Y is the barycentric product of X and Y.

t(X) is the isotomic conjugate of X.

i[X] denotes the isoconjugation fixing X, *i.e.* with pole X&X.

h[Q] denotes Hirst Inversion in C(Q) with pole Q.

If Q = u:v:w and X is x:y:z, then h[Q](X) = vwx^{2} - u^{2}yz : : .

As in the general case, the image of a cubic **K** under h[Q] will contain a cubic precisely when **K**

contains Q and the contacts of the tangents from Q to C(Q). The latter are the intersections of

C(Q) and T(Q) as the latter is the polar of Q in C(Q).

Our first result is proved as in Hirst Inversion - Isocubics. It turns out that, once we know

that the cubic contains the intersections of C(Q) and T(Q), it also contains Q.

**Theorem 1**

Suppose that **K** is one of the cubics pK(W,P) or nK0(W,P).

Then **K** contains the intersections of C(Q) and T(Q) if and only if

(1) P = Q and W is on T(Q&Q), or

(2) W = Q^{2} and P is on T(Q), or

(3) P ≠ Q, W ≠ Q^{2}, and W = Q&h[Q](P).

(equivalently, P = t(Q)&h[Q&Q](W)).

When any of the conditions hold, Q lies on the cubic.

**Theorem 2 - cases (1) and (2).**

Suppose that W is on T(Q&Q).

(1) nK0(W,Q) is the union of T(Q) and C(W&t(Q)).

(2) h[Q](nK0(W,Q)) = nK(Q&Q,W&t(Q))

W&t(Q) is on T(Q).

nK(Q&Q,W&t(Q)) is the image of C(W&t(Q)) under h[Q].

nK(Q&Q,W&t(Q)) is cK(#Q,W&t(Q)) - see notes under CL030.

(3) pK(W,Q) and pK(Q&Q,W&t(Q)) are interchanged by h[Q].

**Theorem 3 - case (3).**

Suppose that P ≠ Q. Let **K** denote nK0 or pK.

Then h[Q](**K**(Q&h[Q](P),P)) = **K**(Q&P,h[Q](P)).

**Corollary 4**

The cubics nK0(W,P), pK(W,P) are invariant under h[Q] if and only if

P is on C(Q) and W = Q&P (*i.e.* W is on C(Q&Q)).

One way to obtain suitable pairs {W,P} is to begin a point X on the line T(t(Q)).

Then take P as the isotomic conjugate of X, and W as the Q-isoconjugate of X.

The families of cubics in Theorem 1 are also closed under another mapping, i[Q].

We begin with a result about the effect of i[Q] on cubics of type nK0 and pK.

**Lemma A**

Let **K** denote nK0 or pK. Then i[Q](**K**(W,P)) = **K**(i[Q&Q](W),P&i[Q](W)).

Observe that C(Q) and T(Q) are interchanged by i[Q]. It follows that, if **K** contains the intersections

of these - so is as in Theorem 1 - then so does i[Q](**K**), so that this is also of the type in Theorem 1.

**Theorem 5**

(1) nK0(Q&Q,P) and pK(Q&Q,P) are invariant under i[Q].

(2) If W is on T(Q&Q), i[Q](nK0(W,Q)) and i[Q](pK(W,Q)) are invariant under h[Q].

(3) If W = Q&h[Q](P), then i[Q](nK0(W,P)) and i[Q](pK(W,P)) are of the type in Theorem 1.

By algebraic computation, we can also establish a result which shows that the operations i[Q] and h[Q]

applied to one of our isocubics produce sets of just six cubics in general, and just three when we start

with a cubic of type (1) or (2) in Theorem 1, or of the type in Corollary 4.

**Lemma B**

The (inverse) maps i[Q]°h[Q] and h[Q]°i[Q] are of order 3.

To illustrate the set of six cubics, we note that, if P is the pivot of a pK (or root of an nK0), then the pole

W is determined as Q&h[Q](P). Thus it sufficient to list the values of P.

cubic | pivot/root |

K |
P |

h[Q](K) |
h[Q](P) |

i[Q](K) |
i[q]°h[Q]°i[Q](P) |

h[Q]°i[Q](K) |
i[q]°h[Q](P) |

i[q]°h[Q](K) |
h[Q]°i[Q](P) |

i[q]°h[Q]°i[Q](K) |
i[Q](P) |

**Examples**

There appear to be few examples of the above types in the current CTC, apart
from those for Q = G

discussed in Hirst Inversion - Isocubics.

The main source is mentioned under CL030.
The cubics cK(#F,R) with R on T(F) are of type cK0. In

the current context, these are related to Hirst inversion in C(F).

**Q = X1**

Then i[Q] is isogonal conjugation. The conic C(Q) has perspector X1, and centre X9.

Those of type cK0(#X1,R) have R on the Antiorthic Axis.

Examples
K040
K137
K221.

**Q = X6**

The conic C(Q) is the Circumcircle

Those of type cK0(#X6,R) have R on the Lemoine Axis.

Examples
K222
K223
K224
K225.

One sporadic example is K367
= pK(X669,X6), with Q = X6, as X669 is on T(X32) (as well as T(X6)).

The image under i(X6) is pK(X1576, X110), and this must be invariant under h[X6]. Of course, we now

know that nK0(X1576, X110) must also be invariant under h[X6]. There is a general pattern for pairs

(W,P) with pK(W,P), nK0(W,P) h[X6]-invariant. If X is on the de Longchamps Axis, then we have the

pair W the isogonal conjugate of X, P the isotomic conjugate of X. Examples culled from ETC are

(X110, X99), (X692, X100), (X1576, X110), (X1492, X789).

**Other Q**

There are three further cases :

K147 = cK0(#X110,X6). T(Q) the Brocard Axis.

K217 = cK0(#X523,X1640). This and next have T(Q) = X115-X125.

K218 = cK0(#X523,X1648).