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 Kx^{2}+Ly^{2}+Mz^{2} = 0, and P = p:q:r, then
P2 = Kp^{3}:Lq^{3}:Lr^{3}.

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(v^{2}/q^{2}-w^{2}/r^{2})]. 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(q^{2}/v^{2}-r^{2}/w^{2})], 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(v^{2}/q^{2}-w^{2}/r^{2})^{2}], on I(Q),

Q3 = [p(q^{2}/v^{2}-r^{2}/w^{2})^{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