The tripolar centroid was introduced in the tucker cubic page. There, we
defined the tripolar centroid of U = [u,v,w] as gU = [u(v-w)(v+w-2u)].

We noted that a circumhyperbola H not through G = X(2) contains two points
with the property that gX is the perspector of H. Each of these gives rise to
four points on H, including the two infinite points. We now have six points
on H.

Here, we identify these points geometrically and indicate some relationships
amongst them.

Definitions
Suppose that H is a circumhyperbola not through G, with perspector P, centre C.
A₁, A₂ are the asymptotes of H,
T₁, T₂ are the tripoles of A₁, A₂.
I₁, I₂ are the infinite points on H, with I_i on A_i,
G₁, G₂ are the second intersections of H with lines through G parallel to the A_i,
P₁, P₂ are the second intersections of H with lines through P parallel to the A_i.

From earlier results, we know that P = gG₁ = gG₂.
We also know that the tripoles of A₁, A₂ are isotomic conjugates.

Observation 1
The circumhyperbola with infinite points I₁, I₁has
(1) P = I₁*I₂,
(2) A₁, A₂ with tripoles I₁/I₂, I₂/I₁.

For points U, V, we have written U*V for the barycentric product of U, V,
and U/V for the barycentric quotient, i.e. U*(tV).

Justification
The circumconic with this perspector contains the points I₁, I₂
The circumconic with these asymptotes has infinite points I₁, I₂
as these lie on the lines A₁, A₂.

The following results are all easy to verify algebraically.

Given A₁

Suppose that A₁ is ux+vy+wz = 0, and put U = [u,v,w].
Then A₂ is x/u+y/v+z/w = 0, the tripolar of U.
Note that the second points are obtained by replacing [u,v,w] by [1/u,1/v,1/w].

P = [u(v-w)2]	this is I1*I2
I1 = [v-w], I2 = [u(v-w)]	these are trival as we know the asymptotes
T1 = [1/u], T2 = [u]	these are trival as we know the asymptotes
G1 = [u(v-w)/(v+w-2u)], G2 = [(v-w)/(uv+uw-2vw)]	the line GI1 is tripolar of [1/(v+w-2u)]
P1 = [u(v-w)2/(uv+uw-2vw)], P2 = [u(v-w)2/(v+w-2u)]	the line PI1 is the tripolar of G2

Observation 2
(1) I₁*G₁ = P₂, I₂*G₂ = P₁.
(2) PI₁ is the tripolar of G₂, PI₂ is the tripolar of G₁.

Given G₁

Suppose that G₁ = [f,g,h]. As indicated above, the infinite points are, say,
I₁ = [g+h-2f] and I₂ = [f(g-h)].

P = [f(g-h)(g+h-2f)]	this is I1*I2 and gG1
I1 = [g+h-2f], I2 = [f(g-h))]	this is the choice above
T1 = [(g+h-2f)/f(g-h)], T2 = [f(g-h)/(g+h-2f))]	as a result of earlier observations
G1 = [f], G2 = [(g+h-2f)/(fg+fh-2gh)]	G1 given, G2 as tripole of PI1.
P1 = [f(g-h)(g+h-2f)/(fg+fh-2gh)], P2 = [f(g+h-2f)]	P1 as I2G2, P2 as I1*G1

In earlier work, we showed that, in the present notation,

(a) the mid-point of G₁G₂ lies on the diameter GC,
(b) the tangents at G₁, G₂ meet on GC.

In fact, these are merely instance of some general results.
We can sum these up as

Theorem 1
Given U,V not on H or equal to C, define U₁, U₂, V₁, V₂ as for G,P.
(a) the mid-point of U₁U₂ lies on UC,
(b) the pole of U₁U₂ lies on UC,
(c) the polar of U is parallel to U₁U₂,
(c) UU₁U₂, VV₂V₁ are in perspective.

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