triangle centres Harmonic homologies with axis the tripole of the centre

Of course, these could be studied as the projective analogue of Bernard's CL012.
Here, we add some further observations. There appear to be a large number of
concurrences and collinearities, some of which are not listed on Bernard's page.

The theory is a special case of that of non-pivotal conics with given centre and axis.
This is discussed in non-pivotal conics. We shall use some early results from there.

notation
For points P, Q :
T(P) is the trilinear polar of P,
C(P) is the circumconic with perspector P,
I(P) is the inconic with perspector P,
C(P,Q) is the nine-point conic of P and Q,
P&Q is the barycentric product of P and Q,
x(P,Q) is the crosspoint of P and Q.

Note : I prefer to thinK0 of the last as the pole of T(P) in I(Q) (or vice versa).

We consider non-pivotal isocubics which are invariant under h, the harmonic homology
with centre X and axis T(X), the tripolar of X. Of course, any such nK0(W,R,X) will contain
the points A' = h(A), B' = h(B), C' = h(C). Elementary projective geometry shows that
these are the second intersections of the cevians of X with the circumconic for which X
is the polar of T(X). This is C(X). These are on all of our cubics. Each example will also
contain the W-isoconjugates of these points. These will lie on a line identified later.

some facts, not all fully digested
Here, we list some facts which were initially observed in Cabri, but then proved by Maple
(or by hand for the very simple ones).

Suppose we choose R on T(X). Then the polar of R in I(X) meets T(X) at a point U.
Equally, U is the pole of RX in I(X) as the polar of X is T(X). Then we have the
non-pivotal cubic nK0(X*U,R,X) with harmonic homology h. Note that we then have
the fact that R is the pole of UX in I(X), by La Hire.

Theorem 1
(1) These cubics occur in pairs nK0(X*U,R,X), nK0(X*R,U,X).
(2) The pair share their inflectional tangent - the line L0 through x(U,X),x(U,R) and X.

The first is simply the observation that the relation between U and R is symmetric.
The second needs a little computation. We will meet the line L0 again.

Note : we now know all the intersections of these cubics - the vertices A,B,C, the three
points A',B',C' mentioned above, and the triple intersection at X.

We now introduce two points which occur on several objects.
R2 is the intersection of RX and T(U),
U2 is the intersection of UX and T(R).

Theorem 2
(1) T(R) is tangent to I(X) at x(R,X).
(2) T(U) is tangent to I(X) at x(U,X).
(3) The tripolars T(R), T(U) meet at a point Y on the line T(X).
(4) T(Y) is the tangent to I(X) at x(R,U).
(5) T(Y) contains R2 and U2.

All are easy to verify by coordinates (or by duality). The point Y occurs again.

Theorem 3
(1) The circumconic through R and U passes through X.
(2) Y is the perspector of the circumconic through R and U.
(3) Y is the pole of L0 in both C(X) and I(X).

Theorem 4
(1) C(X,R) touches T(U) at R2,
C(x(R,X)) and C(X,R)
(2) touch at R, with common tangent T(X).
(3) meet again at two points on T(R).

This is general nine-point conic theory.

Theorem 5
The circumconic C(U)
(1) touches RX at X,
(2) cuts T(X) at points U',U" which lie on nK0(X*U,R,X).
(3) The pole of T(X) in C(U) is x(U,X).

The tangents to nK0(X&U,R,X) at U,U',U" meet at X.

Theorem 6
The circumconic C(x(R,U))
(1) touches T(X) at Y,
(2) meets C(x(R,X)) at R2,
(3) meets C(x(U,X)) at U2.

Note that we now have :
R2 on RX, T(U), T(Y), C(x(R,X)) and C(x(R,U)), and
U2 on UX, T(R), T(Y), C(x(U,X)) and C(x(R,U)).

Theorem 7
The X&U-isoconjugates of the points A',B',C' lie on nK0(X&U,R,X).
They also lie on T(U).

Theorem 8
I(X) and C(R,X) meet at the verices of the cevian triangle of X, and at x(R,U).

Theorem 9 The inflectional tangent contains the intersection of C(U) and C(X),
and that of C(R) and C(X).