Harmonic definitions of a non-pivotal isocubic

Throughout this page, W will denote a fixed point, it will be the pole of our isocubics.
For points X,Y,
X* denotes the W-isoconjugate of X,
T(X) denotes the tripolar of X,
C(X) denotes the circumconic with perspector X,
x(X,Y) denotes the crosspoint of X and Y.

Definition
Suppose that W is a point and L1, L2 are distinct lines, then
K(W,L1,L2) = {M : MM* cut harmonically by L1 and L2 },

Theorem 1
For a point W, and distinct lines L1 = T(X), L2 = T(Y). Then
(1) K(W,L1,L2) = nK(W,R,P), where
R is the x(X,U),
P is the intersection of L1 and L2
(2) L1, L2 each cut the locus again in an W-isoconjugate pair,
(3) L1, L2 cut the T(R) at points on C(x(P,R)),
(4) The cubic is of type nK0 if and only if X* is on L2 (or, equivalently, U* on L1),
Another equivalent that W is on T(X&Y), X&Y is the barycentric product of X and Y.
(5) The locus is invariant under an harmonic homology with axis Li if and only if P* is on Lj.
In a such case, P* is the centre.

Note : (3) is equivalent to meeting on the the ninepoint conic of P and R, by general ninepoint theory.

Proof notes - see harmonic.mws
(1) With Maple, it is easy to compute an equation for the locus. It is a non-pivotal isocubic with pole
W and root R = x(X,U). From the locus definition, it is clear that it contains P.
(2) is another Maple calculation. The W-isoconjugate points on L1 are its intersections with C(X*),
the W-isoconjugate of L1. It turns out that these are also the intersections of L1 with the cubic
other than P. The case for L2 follows by algebraic symmetry.
(3) is a further calculation since we have coordinates for R, P in terms of those of X, U.
(4) follows at once from the equation - we require that the xyz-term vanishes.
(5) requires a different approach. We start with the a W-isoconjugate pair {M,M*} on a line L2.
This determines U, and hence X as a cross conjugate from U and R. The condition that M* is on L1,
and hence is our point P, is exactly the same as that for M to be a flex.

The argument for (5) suggests the following

Corollary 1.1
Suppose that none of W, R, M lie on a sideline. Let K = nK(W,R,M).
There exists a unique point X such that K = K(W,MM*,T(X)).

Proof notes
Suppose that MM* = T(U). As in the previous proof, we identify a point X so that R = x(X,U).
Then T(U),T(X) define the cubic nK(W,R,P) with P the intersection of T(U) and T(X).
By Theorem 1, T(U) meets this cubic in a W-isoconjugate pair. This must be {M,M*} as any
line contains only one such pair. Thus nK(W,R,P) = nK(W,R,M) = K.

There is also a corollary related to Theorem 1 (3). This requires a preliminary result.

Lemma
Suppose that P, R are points, neither on a sideline, with P not on T(R).There exists a unique pair
of points {X,U} such that P is on T(X) and on T(U), and R = x(X,U). X,U need not be real.
The lines T(X), T(U) pass through the intersections of T(R) and C(x(P,R)).

Proof notes
If R = r:s:t, then a typical point on T(R) has the form M(α) = r:αs:-(1+α)t.
Then x(M(α),M(β)) = R if and only if β is a certain linear function of α and α satisfies a
certain quadratic equation. The latter is also the condition for M(α) to lie on C(x(P,R)).
The discriminant of the quadratic vanishes only if P or R is on a sideline, or P on T(R).
Thus we have a unique pair {X,U}, the tripoles of the lines joining P to these points on T(R).

Corollary 1.2
Suppose that none of W, R, P lie on a sideline. Let K = nK(W,R,P).
Then there exists a unique pair of points {X,U} such that P is on T(X) and on T(U), and
K = nK(W,TX),T(U)).

Proof notes
Let X,U be as in the Lemma. Then K = nK(W,R,P) by Theorem 1.

Summing up, suppose that M is a point on a non-pivotal cubic with pole W, root R, but M is not
on a sideline (and hence not on T(R)). Then we can describe the cubic as K(W,L1,L2) with either
(a) L1 = MM*, or
(b) M the intersection of L1 and L2.
In case (b), we have determined the lines L1, L2 in the Lemma. This also gave uniqueness. We
can also identify the lines by the following

Observation
Suppose that K = nK(W,R,M). There are three lines through M which meet K in W-isoconjugate
pairs. These are determined by the intersections of K with pK(W,M) (other than A, B, C).

Proof notes
This is more or less obvious. M is on NN* if and only if N (and so N*) is on pK(W,M).
This cubic meets K in A, B, C and six other points. Thes occur in W-isoconjugate pairs
since the cubics have common pole W. One pair is {M,M*}. Thus we have just two other lines.

Note After Theorem 1 and Corollary 1.2, the lines other than MM* identified in the Observation
define nK(W,R,M) as a locus.