Bernard Gibert's generalisation of tripolar centre see May25.mws

Definition
For R = [r,s,t] ≠ G, the tripolar R-centre of X = [x,y,z] is GR(X) = rx(y-z)(s(w-u)-t(u-v)).

Observe that this is the perspector of the circumconic through X and Y, the intersection
of GX and T(R). Y is r(s(w-u)-t(u-v)).

Also, G_R(X) = {q(w-u)-r(u-v)}/{qv(w-u)-rw(u-v)} gives the same circumconic.

If we take a circumconic C(P) meeting T(R) at P1,P2, then we can recover the points X
with GR(X) = P as the intersections of GP1, GP2 with C(P).

Theorem 1
(1) X = G_R(X) if and only if X is on cK(#G,R).
(2) cK(#G,R) = {X : C(GR(X)) touches T(R)}.

This is a simple Maple check. For the second part, note that, for X on cK(#G,R), the points
coincide, so C(GR(X)) touches the tripolar T(R) at X.

Degeneracy

The "cubic" cK(#G, R) is degenerate if 1/R is at infinity.
Then G is on T(R), and GR(X) is undefined for X on T(R).

Further observations

If U is on T(R), the circumconic C*(U) has perspector P = u2/r on I(R).
It is easy to see this, as U is on T(R), U is on C*(U), with tangent T(R).
Also, 1/U is on C(1/R) - the dual of I(R) - so T(U) is tangent to I(R) at P.

The point on cK(#G,R) corresponding to U is the intersection of GU and C*(U).
This is U* = u/r(v-w) since it clearly lies on the tripolar and conic.

The point U** = r(v-w) is also on T(R) and gives r(v-w)/u, the isotome of U*.

Nodal tangents

U and U** as above are on T(R) and are equal precisely when U is on the diagonal conic
with centre R. This is D(R) : x2/r+y2/s+z2/t = 0. Indeed, the nodal tangents are precisely
the tangents from G to D(R) as the polar of G is T(R).

The best known are R =
X(99) (Steiner),
1/R at infinity - degenerate
X(100) (Jerabek),
D(R) meets T(R) (IK) at I,M (X(9)) so nodal tangents of cK(#G,X(100)) are IG, GM.
The cubic contains X(10), X(58).
X(101) (Kiepert),
Intersections with T(R) unknown.
X(110) (Stammler)
D(R) meets T(R) (Brocard) at O,K so nodal tangents of cK(#G,X(110)) are Euler and GK.
The cubic contains X(251), its isotome, X(97), X(324).

A further calculation shows that, if U,V on T(R) and D(R), then U,V are G-Ceva conjugate.
Summary of work:
U,V on T(R) determine s,t in terms of r.
U,V on D(R) determine s,t in terms of r.
These coincide if and only if V = G-Ceva conjugate of U.

Theorem
The nodal tangents of cK(#G,R) meet T(R) at its G-Ceva conjuagte points.

By applying a projective transformation:

Theorem
The nodal tangents of cK(#F,R) meet T(R) at its F-Ceva conjuagte points.

The diagonal conic is x2/rf+y2/sg+z2/th = 0.

We can identify the cubic if we know one of these intersections :
Given U, let V be its F-Ceva conjugate. T(R) = UV,
so if U = [u,v,w], F = [f,g,h], then R = u/(v/g-w/h).

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