Proofs to follow

Fact 1
If F is not on a line L, there is a unique pair {U,U'} of F-Ceva conjugates on L.
L then also contains [u²/f], [u(v/g+w/h)], [u'²/f], [u'(v'/g+w'/h)].

notes
We are interested in L as T(R). For U = [u,v,w] to be such a point requires
(1) Σu/r = 0,
(2) Σu(v/g+w/h-u/f)/r = 0.
If we add (u/f+v/g+w/h) times (1) to (2), we get
(3) Σ(1/gr+1/fs)uv = 0.
This is the circumconic with perspector [1/gt+1/hs] = [r(s/g+t/h)].
This allows easy construction of the points U, U'.

For F = G this perspector is the complement of tR, so is the centre of I(R).

For cK0(#F,R) - R on T(F) - the perspector is [r²/f] - the R²-isoconjugate of F.

If we subtract, we get the diagonal conic (which appears later)
(4) Σx²/fr = 0.

Definition fX = [f²/x] is the F²-isoconjugate of X.

Fact 2
Let U, U' be the F-Ceva conjugates on T(R).
Then FU, FU' are the nodal tangents of cK(#F,R).
If X, fX lie on cK(#F,R), then FX, FfX meet T(R) in points harmonic with U,U'.

Thus, we have a harmonic pencil FU,FU',FX,FfX.

notes see text
If U on T(R) gives X on cubic then U+ = [r(v/g-w/h)] gives fX.
We get F - nodal tangent FU - when U+ = U.
This is equivalent to U on the diagonal conic (4).
But then we have (4) and (1).
multiplying (1) by (Σu/f) and subtracting 2 times (4), we get
(5) Σu(v/g+w/h-u/f)/r = 0.
So that the F-Ceva conjugate of U on T(R) is on T(R) if and only if U satisfies (1), (4).
The F-Ceva conjugate has its conjugate - U - on T(R), so is the other root of (1),(4).

The last bit is a tedious calculation write a typical point W on T(R) as mU+nV,
where V is the F-Ceva conjugate of U. Then check that W+ is mU-nV.
This uses the fact that, for the choice of U on a nodal tangent, U = U+.

Fact 3
If L, L' are distinct lines through F, then there is a unique point U on L so that
U', the F-Ceva conjugate of U lies on L'.
If L : ax+by+cz = 0, L' : Ax+By+Cz = 0, then
U = [fa(bC+cB)],
U' = [fA(bC+cB)],
UU' has tripole [bC+cB].

This means that we have

Theorem

Given two lines L,L' through F, there is a unique cK(#F,R) with nodal tangents L,L'.

Definition
Given U,V on a line not through F, The F-harmonic of U,U' is the point
W = [uΣu'/f+u'Σu/f] such that (U,U',W,W') = -1, where W' = UU'nT(F).

Once we define the generalised Gibert-tripolar centre as
TFR(X) = r(x/f)(y/g-z/h)(q(z/h-x/f)/g-r(x/r-y/g)/h)
           = rx(yh-zg)(s(zf-xh)-t(xg-yf)).
we get

Fact 4
cK(#F,R) = {P : C(TFR(P)) touches T(R)}.
              = {P : TFR(P) is on I(R)}.

Fact 5 june1.mws
If X is on cK(#F,R) then the line joining the perspectors TFR(X) and TFR(fX)
passes through R+ =[r(s/g+t/h)], the pole of T(F) with respect to I(R).
R+ is the F-harmonic of the perspectors