This generalises the concept of inversion in a circle. We begin with a general version.

Definition
Given a conic C and a point P, the (C,P)-Hirst inverse of X ≠ P is the intersection of the
polar of X and the line PX.

This is clearly an involution which fixes points on C.

We usually take C as the circumconic with perspector P. Then we talk of the point as
the P-Hirst inverse of X, and write it as P(X). Taking P as the centre might be more
natural, but our choice gives a projective configuration.

In barycentric coordinates, if P = [p,q,r], X = [x,y,z], P(X) = [qrx²-p²yz].

A Maple calculation shows the following:
Let K be the cubic nK0(W,R) : Σrx(h²y²+g²z²) = 0, where W = [f²,g²,h²], R = [r,s,t].
Let F = [f,g,h]. Let R be the circumconic with perspector R.
Suppose that Σr/f = 0. Then, if X is on K, F(X) is on R.

Note that the condition actually forces K = cK0(#F,R). Indeed, from earlier work, the
cubic K is obtained by Simson lines from the conic R and U = [r²/f].

Theorem 1
The following are equivalent :
(a) cK(#F,R) is an nK0,
(b) cK(#F,R) is the F-Hirst inverse of R, the circumconic with perspector R,
(c) cK(#F,R) is the locus of tripoles of Simson lines from the circumconic R,
(d) R is on the tripolar of F,
(e) F is on the circumconic R.
(f) The tripolar of F touches the pivotal conic.
(g) For X on cK(#F,R) the F-harmonic of X and its isoconjugate lie on a line.
     The line is the tripolar of the isoconjugate of R and the tangent to R at F.

Notes.
In (c), the condition says that the required conic is actually R.
For details of (f),(g) see the mid-points page.
We also know from TCCT that the cubic is the locus of points X such that
the pole of XX* in the circumconic ABCXX* lies on a line L, where X* is the
isoconjugate of X. This is the same line as in (g), but the map from X,X* to
the line is different. Here, we associate with X,X* the F-harmonic of the meet
of XX* with the tripolar of F. Thus the condition may be rephrased as

The cubic is the locus of points X such that the pole of L with respect to the
circumconic ABCXX* lies on the tripolar of F.

Theorem 2
In the above notation, suppose that F-Hirst inversion sends conic R to cubic K with node F.
Let U be the F-Ceva conjugate of R. For X on R, let X' the second meet of R and UF.
Then the F-Hirst inverses of X and X' are F-isoconjugate.

This is purely projective - it then follows from the known case K185. There U can be described
in projective terms as the pole with respect to the Kiepert hyperbola of the tripolar of F. This
generalises to the point U described in the Simson page.
Indeed, the two-point theorem for central conics shows that the point U is the harmonic of the
intersection of XX' with the polar of U (the tripolar of F) with respect to X,X' for all X.

Examples include :
K040 with F = I, R = X(650), R the Feuerbach hyperbola, F the circumconic perspector I.
Here U = [a(b-c)²(b+c-a)²] which appears to be unlisted.

K185 with F = G, R = X(523), R the Kiepert hyperbola, F the Steiner Ellipse.
Here U = X(115), the centre of the Kiepert hyperbola.

Both cubics also have the mid-point property - the cubic is the locus of points X such that
the mid-point of X and iF(X) lie on a line. The line is Σr'x = 0, where R' = [r',s',t'] is the
anticomplement of R.

The condition for this property is that Σr'f = 0, i.e. F is on the line. Equivalently Σf'r = 0.

For F ≠ G, both conditions hold only when R = [f(g-h)(g+h-f)].
The mid-point line contains both F and its barycentric square F².
This occurs since the anticomplement of R is then [(g-h)/f].

When F = I, this is K040. The mid-point line is IK.

When F = K, we get a new cubic:

L001

F = K, R = X(647), R is the Jerabek Hyperbola, F is the Circumcircle.
The mid-point line is the Brocard axis.
Here, U = [a²(b²-c²)²(b²+c²-a²)²] is unlisted.
As O is on R, its K-Hirst inverse X(511) lies on the cubic.
X(511) is the infinite point of the Brocard axis.
The isoconjugate X(1976) must also on lie the cubic.
Now the line OU must meet R again at the K-Hirst inverse of X(1976).
The circumcircle and the Jerabek hyperbola meet at X(74), so this is on the cubic.
Its isoconjugate is X(1495), so this lies on the cubic.
The line of F-harmonics is therefore KX(1495), i.e. the line KX(25). This will be the
tangent to the Jerabek hyperbola at K.

The nodal tangents are determined by the intersections of the Jerabek hyperbola and
the Lemoine axis.

Note that any case with F = K will have R on the Lemoine axis by (d) of the Theorem.

In general, the infinite point on the mid-point line must lie on the cubic as the "mid-point"
of a point Q at infinity and its isoconjugate. As Q is at infinity, this is Q.

For F = G, the conditions are both equivalent to "R on the line at infinity".
Then the line is Σrx = 0, the tripolar of the isotomic conjugate of R.
When R = X(523), the line is GK. The point U above is X(115).
In general, as U = [r²], U is on the Steiner Inellipse; it is the centre of R.

new menu