- Oct 3, 2005Wilson,

On Oct 2, 2005, at 12:02 PM, Wilson Stothers wrote:

> I try to explain below why the operation [cross conjugate] is

> natural and ought to be

> included in this theory sooner rather than later.

I do not know these operations, and am initially suspicious of them,

but I should know about them. I tend to think in terms of more

primary operations like isogonal conjugate and medial (perhaps now

supplement) operations. Below you see that this can be effective.

> ...

>

>> The interesting part is that the isogonals of sP and sgP are

>>

> collinear

>

>> to P and gP!

>>

>

> Very nice - I had not noticed this!

I think that this is a grand structural principle, it is true for

other coordinate systems as well, such as barycentrics. If you think

interms of symmetric functions (easy with barycentrics but not with

trilinears) the positions of all centers with known coordinates can

be derived from knowing them. In short I am very impressed with this

collinarity.

>

>

>> We now have 4 points in the P�gP line. Their tripolars meet at the

>> perspector of the circumconic isogonal to this line with P, gP, sP,

>> and sgP being on this conic .

>>

>

> I think you mean the tripolars of the initial points P,gP,sP,sgP,

> rather than those of their isogonals.

yes

>

> Some further thoughts:

>

> Familiar operations can be derived from s,g and a( the inverse of s):

>

> gsP = cevapoint of I, P,

> sgP = crosspoint of I, P,

> agP = P-ceva conjugate of I,

> gaP = P-cross conjugate of I.

Operations not familiar to me. I will probably stick to g, s, a. I

like simplicity. No trinary operations when binary or unary ones will

do. Relationships expressed with as few operations as possible. This

is my prejudice, but I am happy if you can convince me otherwise.

>

> Algebraic symmetry suggests the inclusion of the cross-conjugate

> since we have inverse pairs gs,ag and ga,sg.

>

> Conversely, if we know about Q-ceva conjugates and crosspoints:

>

> aP = gP-ceva conjugate of I, so aP is the cevian quotient gP/I,

> sP = crosspoint of gP and I.

>

> Geometrically, we have collinearities including

>

> I,P,sP,aP, and, replacing P by gP,

> I,gP,sgP,agP,

>

> and Steve's example

> P, gsP, gP, gsgP

>

> But there is further geometry which prompts two further operations.

>

> We write

>

> Cir(Q) for the circumconic with perspector Q,

> Inc(Q) for the inconic with perspector Q.

>

> gsP

> = tripole of polar of P in Cir(I)

> = tripole of polar of I in Cir(P)

> sgP

> = pole of tripolar [tripole I assume] of P in Inc(I)

> = pole of tripolar of I in Inc(P)

> agP

> = pole of tripolar of P in Cir(I)

> gaP

> = tripole of polar of P in Inc(I)

Now this is interesting, very interesting perhaps.

>

> It now appears that there are two "missing" operations:

>

> xP = pole of tripolar of I in Cir(P) (the I-ceva conjugate of P)

> yP = tripole of polar of I in Inc(P) (the I-cross conjugate of P)

>

> I was surprised to find that these can be expressed using the

> basic operations a, s and g

>

> xP = sgaP

> yP = gsgagP

>

> These make it clear that the operations have order 2.

>

> Also, xP can be regarded as the crosspoint of I and aP,

> or as the supplement of the P-cross conjugate of I.

>

> The operation zP = agsP also has order 2, but I can't see the

> geometry.

> It is, however, ag(sP), the sP-ceva conjugate of I,

> As such, it does make brief appearances in ETC (X1044,..,X1054).

> It is also a(gsP), the antisupplement of the cevapoint of P and I.

>

> I hope this helps with your investigation of supplements and the like.

Very much, and very nice.

but I will stick with the basic operations and leave out the fancy

words for now. Counting a, s, g, t you are using 10 operations here.

I will stick with 3.

Mineur presented this operation as a parallel to the complementary

one for which I long ago worked the analogues to what we have been

doing. I have been surprised how nice it is working. Once you

convinced me that the supplement of P was not supposed to be P with

respect to the incentral tirangle, things have fallen into place. I

think there is a general structure for which I have worked out the

isotomic case (but not the conic relationships you list above and

which interest me a lot!) and we (you mostly) are working out the

isogonal one. I suspect that this structure exists for all conjugations.

I will email you my work on the affine invariant case, which contains

the additional advantage of allowing one to find the positions of

points on lines, but it is otherwise very similar.

Regards,

Steve

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