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 U^{2} = [u^{2}].

This is on SI, and fU^{2} = 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 [(b ^{2}-c^{2})/(b^{2}+c^{2}-2a^{2})] 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 (p^{2}+q^{2}+r^{2}-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**

*
*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)**

**
****
**

**
return to parabolas
**

**
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 = fW**

**
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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