Darij Grinberg introduced the notion of tripolar centroid. See ETC for details.
If U = u:v:w = [u], then the tripolar centroid of U is TG(U) = [u(v-w)(v+w-2u)].

In fact, the point TG(U) has a significance in the theory of circumconics.
The circumconic C(U) with perspector TG(U) clearly passes through the
points U, V = [u(v-w)], W = [(v+w-2u)], and X = [u(v+w-2u)].

We can also check that T = [u/(v-w)²] is on C(U).

C(U) is the isotomic conjugate of the line L(U) whose coefficients are the
coordinates of TG(U).

V and W are at infinity, so C(U) has points at infinity. It is not an ellipse.
C(U) will be a parabola when V,W coincide. We shall consider this below.
For the moment, we will assume that they are distinct, so that C(U) is a
hyperbola, and V,W are the infinite points on the asymptotes.

Now, V is the infinite point on the tripolar of U, and W is the infinite point
on the line GU, where G is the centroid of the reference triangle. We have

(1) C(U) passes through U,
(2) C(U) has an asymptote parallel to the tripolar of U,
(3) C(U) has an asymptote parallel to GU.

Of course, any two of these specify C(U) uniquely.

To cope with the degenerate case, we may rephrase (2), (3) as

(2') C(U) passes through the infinite point of the tripolar of U.
(3') C(U) passes through the infinite point of GU.

Further, X lies of the tripolar of U, so is the finite intersection of this line with
the circumconic.

Also, since we know the equation of C(U) and the coordinates of the infinite
points, we can write down equations for the asymptotes. They are the tripolars
of U* =[u(v-w)/(v+w-2u)] and its isotomic conjugate.

Finally, the fourth intersection of C(U) with the Steiner ellipse is the isotomic
of Y, the infinite point of L(U). A Maple calculation shows that Y is the point
[2u(v²+w²) - v(w²+u²) - w(u²+v²)].

examples

We know that, if U = X(6) = [a²], then TG(U) = X(351), the centre of the Parry
circle. Then V = X(512), W = X(524), X = X(187), T = X(249).

C(U) is the isotomic conjugate of the line X(76)X(338), so passes through
X(249), X(598).

By looking at the isogonal conjugates of X(6)X(512), we see that C(U)
is the isogonal conjugate of the line X(2)X(99). It therefore passes through
X(843), which lies on the Circumcircle.

Here the tripolar of U is the Lemoine axis, so one asymptote is parallel to this.
The other asymptote is parallel to GU = X(2)X(6), the major axis of the Lemoine
inellipse.

If U = X(1), then TG(U) = X1635). Here, V = X(513), W = X(519), X = X(44),
T = X(765).
Looking at the isogonal conjugates, C(U) also passes through X(679), X(751),
X(765), X(1168), X(1319). Looking at isotomic conjugates, we get no new points.
Here the asymptotes are the tripolars of X(1022) and its (unnamed) isotomic
conjugate. We observe that TG(X(1022)) = X(244). This is the perspector of
the circumhyperbola with asymptotes the tripolars of X(1) and X(75).

degenerate cases

We now look at the case where C(U) is a parabola. This will occur if and only if
the perspector TG(U) lies on the Steiner inellipse. Or, equivalently, if and only if
the centre, the G-Ceva conjugate of TG(U), is at infinity. Either way, we have

C(U) is a parabola if and only if U lies on the curve

                    u(v²+w²) + v(w²+u²) + w(u²+v²) - 6uvw = 0.

This is the Tucker Nodal Cubic, usually denoted T(G). It contains no known triangle
centre other than G.

Since C(U) passes through the infinite point of the tripolar of U,

When U is on T(G), the tripolar of U is parallel to the axis of the parabola C(U).

It is known that, if U lies on T(G), then the line joining U and its isotomic conjugate
touches the Steiner ellipse. Some Cabri sketching suggests that the contact point
is the fourth intersection of C(U) and the Steiner ellipse.

Of course, the isotomic conjugate of this line is a circumhyperbola through U
and its isotomic conjugate U'.

In this case, the point Y simplifies to [u(v-w)²] and Y' does lie on UU'.

This allows us to recover the point P for which a given circumparabola is C(P).

Suppose that C is a circumparabola, and that its fourth intersection with the Steiner
ellipse S is the point F. Then C = C(P), where P is the second intersection of C
with the tangent to S at F.

Gibert's website also gives the following fact :

For any P, let Q be the centre of the inconic with perspector P, and let C be the
circumconic through P and Q. Then C is a parabola if and only if P lies on the
Tucker Cubic T(G).

We can also describe C as the circumconic

(a) through P and its isotomic conjugate P',
(b) through P and its complement,
(c) the isotomic of the line PP'.

We refer to it as the Tucker conic T(P) for P.

T(P) must be C(R) for some R, identified as above. We know the equation of T(P).
If P = p:q:r, the fourth intersection of T(P) and S is 1/[(q+r)(p²-qr)]. We can now write
down an equation for the tangent to S at this point. We can now find the intersections
of this tangent with T(P). One is the fourth intersection. The other gives the required R.

If P = p:q:r, is on T(G), then T(P) = C(R), where R = [p(q-r)/(p²-qr)].

R is defined in general, but C(R) contains P and Q = [p(q-r)] only if P is on T(G)
or a certain sextic.

Thus, given P on T(G), we can find one of the two circumparabolas through P by
Gibert's result. Our work has provided another.
If P = p:q:r, then Gibert's point Q is [p(q+r)]. The circumhyperbola has perspector
P1 = [p(q-r)(q+r)]. Our circumparabola has perspector P2 = [p(q-r)(q+r-2p)].
Thus the conics are distinct. Some straight-forward algebra yields :

For any P, define Q,P1,P2 by the formulae above. The mid-point of P1P2
lies on GQ if and only if P lies on the Steiner ellipse or the Tucker Cubic.
When P lies on the Steiner ellipse, the mid-point of P1P2 is G.

Note that a circumconic through a point P has perspector on the tripolar of P.
A circumconic is a parabola if and only if its perspector is on the Steiner inellipse.
Thus there are at most two circumparabolas through a given P.

geometry

We know that the asymptotes of C(U) are parallel to the tripolar of U and GU.
We also know that a line parallel to an asymptote of a hyperbola H meets H
in one finite point. Thus, if H = C(U), then U must be one of the meets of H
with the two lines through G parallel to the asymptotes.

If H = C(U), then the second point is the intersection of H with the line through
G parallel to the tripolar of U. A routine calculation gives the point U*, where
U* = [(v+w-2u)/(uv+uw-2vw)]. We need to check that TG(U*) = TG(U).

We can also recover U* as the isotomic of the intersection of the isotomic of
C(U) and the line joining U to the fourth intersection of C(U) and S.

Or, indeed, as the second intersection of C(U) with the line joining the isotomic
conjugate of U and the fourth intersection of C(U) and S.

geometry of circumparabolas

Suppose that C is a circumparabola.
Then C has perspector P which lies on the Steiner inellipse.
C is the isotomic conjugate of L the tripolar of the isotomic conjugate of P
L is the dual of P, so is a tangent to the Steiner ellipse.
The intersections Q,Q' of L with C are isotomic conjugates as they are
on both L and its isotomic conjugate.

Since C is a parabola, Q,Q' lie on T(G).
Also since C is a parabola, there is a unique point U on T(G) with C = C(U).
As T(G) is self-isotomic, the intersections of C and T(G) are the isotomic
conjugates of the intersections of C and L. We have identified Q.Q' as two
of these. There is a unique third. This gives U as the third intersection of C
and T(G).

C,T(G) have three intersections, two of which are isotomic conjugates.
C = T(Q) = T(Q'), where Q,Q' are the isotomic conjugate intersections.
C = C(U), where U is the third intersection.

Even if T(G) is not available, we can still locate the points Q,Q',U

Q,Q' are the intersections of L and C.
U is the second intersection of C with the tangent to S at the fourth
intersection with C.

Also, given K on T(G), we can find the circumparabolas through K as

C(K), the circumconic with perspector TG(K), the tripolar centroid.
T(K), the circumconic through K and its isotomic conjugate.

some algebra

It may seem likely that a circumconic could be described as C(U) for several
centres U. Then the perspector will be the common value of TG(U). A Maple
calculation with resultants shows that

TG(U) = TG(Z) only for Z = U or for Z = U* = [(v+w-2u)/(uv+uw-2vw)].

We outline the calculation :
Suppose Z = x:y:z has TG(Z) = TG(U). Then the coordinates satisfy equations of
the form e1 = u(v-w)(v+w-2u)-kx(y-z)(y+z-2x) = 0, with e2, e3 defined similarly.
Using resultants, we eliminate k to get expressions e12 and e13.
Another resultant eliminates x, giving an expression e123.
Solving e123 = 0 for z, and scaling, we get (y,z) = (v,w) or those in U* above.
Solving e12 = 0 and e13 = 0 for x, we get two answers. These will agree only
when Z = U or U*.

We know of no examples where U and Z are named centres.

As a corollary, we obtain :

A circumhyperbola contains two points whose tripolars are parallel to asymptotes.

The existence of the two points needs a further calculation. If we write down the
condition for the point R = r:s:t to be a tripolar centroid and use resultants much
as above, then we obtain a quadratic equation which has real solutions if and only
if R lies outside the Steiner inellipse. Thus, the centre of every circumhyperbola is
the tripolar centroid of (two) real points.

The observations above on C(X(1)) generalise easily.

The circumhyperbola with the tripolar of U as asymptote is C(H), where
H = [u(v-w)/(v+w-2u)] or [(v-w)/(uv+uw-2vw)].

Implicit in this are some algebraic identities.

Write f(U) = [(v+w-2u)/(uv+uw-2vw)], g(U) = [u(v-w)/(v+w-2u)]. Then

TG(f(U) = TG(U).
f(f(U)) = U.
g(g(U)) = U.

The pairs giving the same hyperbola may include one point at infinity.
This occurs precisely when G lies on an asymptote.
If U is the finite point of one such pair, then U* = x:y:z lies on the curve :

x³y²+x²y³+x³z²+x²z³+y³z²+y²z³ +2xy²z²+2yx²z²+2zx²y²-4x³yz-4xy³z-4xyz³ = 0.

footnote

Now thxt we know that a circumhyperbola can be described in two ways -
as C(U) and as C(f(U)). This gives six points on the conic:

[u], [(v+w-2u)/(uv+uw-2vw)], the generating points.
[u(v-w)] and [v+w-2u], the infinite points.
[u(v+w-2u)], [u(v-w)(v+w-2u)/(uv+uw-2vw)], the intersections with the tripolars of U, f(U).