Additional notes on the nine-point conic

As a temporary measure, I will use notation of [BC], combined with the Vi, Wi notation
for certain triangle centres on the conic C(P,Q) as in my notes [WWS].

Recall the notations
T = crosspoint of P and Q,
R = cevapoint of P and Q.

We also need another point derived from P and Q in a symmetric way.
W = intersection of PQ with the polar of P in C(Q),
= intersection of PQ with the polar of Q in C(P)
This point obviously lies on PQ.

Clearly, C(P,Q) is a circumconic of the Cevian triangles of each of P and Q.
There are two associated perspectors,
P1 = perspector of C(P,Q) in the Cevian triangle of P,
Q1 = perspector of C(P,Q) in the Cevian triangle of Q.

Translating from [WWS], we have

Theorem 1
P1 is the P-complement of T, so lies on PT. It also lies on W1W4 and W2W3.
X,Y on C(P,Q) are P1-antipodes if and only if their Q-Ceva conjugates are conjugate in I(P).
Q1 is the Q-complement of T, so lies on QT. It also lies on V1V4 and V2V3.
X,Y on C(P,Q) are Q1-antipodes if and only if their P-Ceva conjugates are conjugate in I(Q).

We now add some information which allows us to identify P1 and Q1 directly.

Theorem 2
P1 is the intersection of PT and RW.
Q1 is the intersection of QT and RW.
Also, (R,W,P1,Q1) is harmonic.

P1 = [p(2p/u+q/v+r/w)],
Q1 = [u(2u/p+v/q+w/r)].

The case P = X(2), Q = X(4)

R  = X(264),
W  = X(297),
P1 = X(141),
Q1 = X(53).

The conic C'(P,Q) is the common circumconic of the Anticevian triangles of P and Q.
This exists as it is the diagonal conic through P and Q. As a circumconic, it will have
perspectors with respect to each of these Anticevian triangles.

General Result
If P is on a diagonal conic C, then P2, the perspector of C with respect to the
Anticevian triangle of P, lies on T(P).
If C has equation Kx2+Ly2+Mz2 = 0, and P = p:q:r, then P2 = Kp3:Lq3:Lr3.

The coordinates of P2 were computed by Maple. It is on T(P) as P is on C.

Let P = p:q:r, Q = u:v:w.

We know that Q is also on the conic C'(P,Q). This allows us to compute K,L,M.
P2 = [p(v2/q2-w2/r2)]. This is on T(P), and also on T(P"), where P" is the
P-complement of Q, i.e. P" = [p(v/q+w/r)]. Thus, it is their intersection.

As P2 is on T(P), its Q-Ceva conjugate is on C(P,Q). This is the point W6, given by
W6 = [p((v/q-w/r)2(-u/p+v/q+w/r)].

W6 is on the inconics I(P-anticomplement of Q) and I(Q-cross conjugate of P).

W6 is the P-antipode of W5, the P'-antipode of V4 and the P"-antipode of W3.
P' is the P-complement of P", i.e. the pole of T(P) in C(P,Q).

W6 is on the tripolar of the point P3, where P3 = [p(v/q-w/r)].
This is the intersection of T(P) and T((P,P)-isoconjugate of Q).
As P3 is on T(P), its U-Ceva conjugate is on C(P,Q). It is W2.

Analagously, we have the points
Q2 = [u(q2/v2-r2/w2)], on T(Q), and its P-Ceva conjugate
V6 = [u((q/v-r/w)2(-p/u+q/v+r/w)], on C(P,Q).

Note
These points V6, W6 appear in [WWS], but there the P, Q-Ceva conjugates are
wrong. In addition, the notation there swaps V6 and W6 compared to here,
where we get W6 as a Q-Ceva conjugate. It will also be a P-Ceva conjugate,
but the relevant point on T(Q) does not seem significant. It is the intersection
of T(Q) and T(Q*), where Q* is the P-Ceva conjugate of Q.
Here we have identified the Ceva conjugates geometrically as perspectors.

We now have twelve points {Vi, Wi, i = 1..6} on C(P,Q).
Each can be constructed by straight-edge from P and Q.
If P and Q are triangle centres, each is a triangle centre.

Remark
There are also perspectors associated with the conics
I(P,Q), the inconic of the Cevian triangles of P and Q, a diagonal conic, and
I'(P,Q), the inconic of the Anticevian triangles of P and Q.
These can be found by G(P,Q)-duality (see [WWS2]).
The former gives perspectors
P3 = [u(v2/q2-w2/r2)2], on I(Q),
Q3 = [p(q2/v2-r2/w2)2], on I(P).
The latter gives complicated perspectors which do not have obvious geometry.

[WWS]  nine-point notes
[WWS2] duality
[BC]      Bernard Gibert's Bicevian Conics