results on the tucker cubic

First a new(?) characterization of the Tucker Cubic TC.

Recall that G is the centroid.

Theorem 1
If P ≠ G, then P is on TC if and only if PG is parallel to the tripolar of P.

Proof
If P = [p,q,r], then the infinite point of PG is U =[q+r-2p].
The infinite point of the tripolar of P is V =[p(q-r)].
The result is a simple verification that U = V.

Note. this means that if P is on TC, and F is defined in terms of P,
then in calculation, we can replace "p(q-r)" by "q+r-2p" systematically
in the coordinates of F.

Thinking of the isotomes, another pair is [pq+pr-2qr] and [q-r].

a useful transform

Definition For U = [u,v,w] ≠ G, the f-transform of U is fU = [u/(v-w)].

The f is for fourth - see below.

We can see that fU is the fourth intersection of the conic with perspector U
and the conic with centre U, i.e. the intersection other than the vertices.
Of course, for U = G, these conics coincide.

From the formulae, fU is the barycentric product of U and the tripole of GU.
Again it is clear why we exclude U = G.

If V = [u(v+w-u)], the G-Ceva conjugate of U, then the conics for fU, fV are
the same, so fU = fV. A simple Maple calculation shows that this is the only
case where fX = fU, X ≠ U. Another that we can find U with fX = W if and
only if W lies outside the Steiner Ellipse. W on the ellipse gives X = G.

Theorem 2
(1) fX = fU if and only if X = U or the G-Ceva conjugate of U,
(2) fX = W is solvable if and only if W is outside the Steiner Ellipse.

A simple Maple calculation shows that we have

Theorem 3
fU is on the Tucker Cubic if and only if U is on the Line at Infinity
or the Steiner Inellipse.

Note that, if U is on LI, then the G-Ceva conjugate is U2 = [u2].
This is on SI, and fU2 = fU by Theorem 2(1).

We now have triangle centres other than G on the Tucker Cubic.
Take U at Infinity, then fU is on the Tucker Cubic.

Examples
[(b-c)/(b+c-2a)] and [(b2-c2)/(b2+c2-2a2)] lie on TC.
As do their isotomes. None of these are in the current ETC.

Darij Grinberg's tripolar centroid

In later versions of ETC, the tripolar centroid of a point is defined.

Definition
For U = [u,v,w] ≠ G, the tripolar centroid of U is pU = [u(v-w)(v+w-2u)].
We refer to the map U to pU as the p-transform.

The p is for perspector - see below.

It is a matter of simple observation that pU is the perspector of the circumconic which
passes through U, [u(v+w-2u)], [u(v-w)] and [v+w-2u]. The latter are at infinity, so
the conic cannot be an ellipse.

If we try solving the equation pX = P, P = [p,q,r], X = [x,y,z], Maple gives 2 expressions
which involve the square root of (p2+q2+r2-2pq-2qr-2rp)(p+q+r)2. But if p+q+r then the
solutions degenerate to X = G, when pX is undefined. This corresponds to the perspector
of a Circumconic through G. The other case of equal roots is for P on the Steiner Inellipse.
This is the perspector of a Circumparabola.

Theorem 4
(1) When P is the perspector of a Circumparabola, pX = P has a unique solution
(2) When P is the perspector of a Circumhyperbola not through G, then pX = P
has two solutions.

Although the existence of just two solutions emerges from the algebra, there is an
alternative more geometric argument. We note that pU is the perspector of the
Circumconic through U and [u(v-w)], the infinite point on the tripolar of U.
Suppose we have a Circumconic C with an infinite point W, and perspector pU
with W on the tripolar of U. Then U lies on the Circumconic with perspector W
as well as on C. But there is only one such point other than the vertices. Thus
the number of U with pU the perspector of C is equal to the number of infinite
points on C.

In fact we can find the points in a different way, with a simple geometrical construction.

Theorem 5
Suppose that C, the Circumhyperbola with perspector P, does not pass through G.
The points X on C with pX = P are the intersections of C with the lines through G
parallel to the asymptotes of C.

Proof
Suppose that C has asymptote with tripole U. Then the second asymptote has
tripole tU (from other work).
Then P = [u(v-w)2] as the product of the infinite points of the asymptotes.
Now it is easy to check that X = [v-w)/(2vw-uv-uw)] and Y = [u(v-w)/(2u-v-w)]
lie on C and on the stated lines. The latter since these have equations of the
form kx+ly+mz = 0, where [k] = [2u-v-w], [2vw-uv-uw].
Now, from the coordinates, G is on C if and only if U is on TC.
If we find pX, pY by Maple , these are P (multiplied by the Tucker function).

Maple also shows that:
(1) XY is parallel to the isotome of C,
(2) the mid-point of XY is on GZ, where Z is the centre of C.
(3) the tangents at X and Y meet on GZ.

This means that GZ is the diameter conjugate to that parallel to the isotome
of C (and hence to XY).

Corollary
For U not on TC, pU is the perspector of the Circumconic through U,
with an asymptote parallel to UG.

This could be observed from the original definition.
In fact, this also gives Theorem 5 to the extent that it identifies U as the
second intersection of the conic with a line through G parallel to an asymptote.

But now (1) gives us more.
The second point V with pV=pU is the second intersection with the line through
U parallel to the isotome of C. The second asymptote is parallel to VG

We also have, by a little Maple from Theorem 5,

Theorem 6
For U not on TC, pU is the perspector of the circumhyperbola with
asymptotes the tripolars of [u(v-w)/(v+w-2u)], [(v+w-2u)/(u(v-w))].

Also, if the Circumhyperbola C has perspector pU, then the second infinite
point is [v+w-2u] and the argument will yield the second solution. It is easier
to invoke Maple to get the following result.

Theorem 7
If pX = pU, then X = U or [(v+w-2u)/(uv+uw-2vw)].

another transform

Theorem 6 suggests another transform:

Definition
For U = [u,v,w], dU = [u(v-w)/(v+w-2u)].

The d is for dual - see below.

For all U on TC, dU evaluates to G as then [u(v-w)] = [v+w-2u].

Theorem 6 states that the circumhyperbola with perspector pU has asymptotes
kx+ly+mz = 0 and x/k+l/y+m/z = 0, where K = [k,l,m] = dU. We often refer
to the first of these as the dual of K. The second is the tripolar of K.

Suppose we take the circumhyperbola H with the dual of U as an asymptote.
Then the other asymptote is the tripolar of U. The perspector is [u(v-w)2],
so G is on H if and only if U is on TC. We might ask for the point X on H whose
tripolar centroid is the perspector and which lies on the parallel to the dual of
U through G (see Theorem 5). The answer is easily found by Maple. It is very
surprising. Once known it is easy to verify.

Theorem 8
For U not on TC, the circumhyperbola with perspector p(dU) has as
asymptotes the dual and tripolar of U.
The second point X with pX = p(dU) is X = d(tU).

Proof
A calculation gives the perspector as [u(v-w)]2, as required.
The second part is trivial as each asymptote is the dual of the isotome
of the dual of the first.

Implicit in this is the fact that the transform d has order 2.

Example
Take U = X(1). The circumhyperbola with asymptotes the dual
and tripolar of X(1) has perspector X(244) = [a(b-c)2].
X(244) = pX(1022), X(1022) = [a(b-c)/(b+c-2a)] = dX(1).
tX(1) = X(75), but dX(75) = [(b-c)/(ab+ac-2bc)] is unlisted.
The centre is X(661) = [a(b2-c2] as the G-Ceva conjugate of X(244)
or as the intersection of the asymptotes.
pX(1) = X(1635), so the circumhyperbola with perspector X(1635)
has asymptotes the dual and tripolar of X(1022).

The earlier argument does reveal that if C is a parabola with centre W, then
U must be the intersection of C with the circumconic with perspector W, so it
is fW. Theorem 3 now gives us a contorted proof of the fact that, if C is a
Circumparabola, then its perspector (being fW, W at infinity) is on TC.
Also fW = fW2, so we have

Definition
For a Circumparabola C with perspector P, fP is the third Tucker point of C

It is the third intersection of C with the Tucker Cubic, other than the vertices.
If we look at the istomes, C gives a line L, TC is self-isotomic (ignoring the vertices).
Thus there are at most 3 intersections. Two are the intersections of C and TC, the
isotomic pair on C. We have found the third.

Theorem 9
The f-transform maps the Steiner Inellipse to the Tucker Cubic with G deleted.
It maps U to the third Tucker point of the Circumparabola with perspector U.
The inverse is the p-transform.
It maps V to the perspector of the Circumparabola with third Tucker point V.

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