Re: [EMHL] Re: supplementaire

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• Wilson, Thanks, this is nice. I feel there is finally a subject here. I think the Cevian conjugate is what I call the Cevian quotient. Now I have to figure out
Message 1 of 21 , Sep 28, 2005
Wilson,

Thanks, this is nice. I feel there is finally a subject here.

I think the Cevian conjugate is what I call the Cevian quotient.

Now I have to figure out what the cross-conjugate is, and whether it
is a significant operation.

Steve

On Sep 28, 2005, at 9:32 AM, Wilson Stothers wrote:

> Dear Steve
>
> Can I perhaps add a few more sets of points,
> and fill a few gaps.
>
> Let me write
> sP for the supplement of P,
> aP for the antisupplement of P,
> gP for the isogonal conjugate of P
>
> Function P sP aP
>
>
> sin A 6 37 9
> csc A 2 42 43
>
> tan A 19 48 610
> cot A 63 31 1707
>
> cotA/2 9 6 1743
> tanA/2 57 55 165
>
> These give new and simple forms for the trilinears
> of X610, X1707, X1743, and alternative versions for
> those of X6, X31, X37, X42.
>
> I cannot find much geometry, but do observe that
>
> aP = gP-Ceva conjugate of I
> gP = sp-cross conjugate of I.
>
> Also, (P, sP, aP, I) is harmonic,
> as is (P, aP, sP, X), where
> X is the intersection of IP with the antiorthic axis.
>
> Regards,
>
> Wilson
>
>
>
>
> --- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@c...> wrote:
>
>> If a point has form :y: in trilinear coordinates then
>>
>> :z+x: is its "supplementaire" and :z+x-y: its antisupplementaire.
>>
>> Trilinears seem best to me using angles, so for angle functions
>>
> here
>
>> are some correlations that I have found by searching Kimberling's
>>
> ETC.
>
>>
>> This is part of my mining the ETC project, for which negative
>> correlations are as interesting as positive ones.
>>
>> I would love it is people knew some geometry that went with these
>> combinations.
>>
>> P Io supplementaire
>>
> antisupplementaire
>
>> | function
>>
>> 3, 1, 65,
>> 46 cos B
>> 6 1 --
>> -- sin B
>> 4 1 73
>> 1745 sec B
>> 2 1 --
>> 43 csc B
>> 266 , 1, --,
>> 164 sin B/2 258 on this line
>> 174 1, --,
>> 503 sec B/2
>> 188 1 , --,
>> 361 csc B/2
>> 259 1 ----
>> 173 cos B/2
>>
>
>
>
>
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• Following Wison s lead and using his notation, I think I am getting a handle on this supplement operation. I am interested in this because the classical
Message 2 of 21 , Oct 1, 2005
Following Wison's lead and using his notation, I think I am getting a
handle on this "supplement" operation. I am interested in this
because the classical geometers used it and because it is not
mentioned in ETC, and thereby presumably by modern geometers, the
general principle here being that finding out what is not said is
often as interesting as what is said. I am also interested in points
whose trilinears coordinates are half-angles of A, B,C because they
do not relate well to the great mass of points and because they have
unique extraversion properties.

I have combined Wison's and my tables below for reference.

Some geometry from the supplementaire's.

Consider P and its conjugate (alway isogonal) gP. Considering both
enforces what is called anallagmatic symmetry. More symmetry is better.

sP can be constructed as the the cevian quotient gP/Io by connecting
the ex-incenters to the cevian traces of gP.

Similarly for sgP.

Now get aP and agP as harmonic conjugate (as Wison explains below).

I am ignoring the cross conjugate operation because it seems too
complicated for me to understand it as other than a mathematical
operation. I do notice that it is an involution (to what?) and that
may make it interesting later.

The interesting part is that the isogonals of sP and sgP are colinear
to P and gP!

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 . For example if P = H, the orthocenter,
this conic is the isogonal of the Euler line, the Jerabek conic. From
our lists below, we find out that X(65) and X(73) are on this conic.

The Jerabek conic is unique in that one can do this same game in
barycentrics using isotomic conjugation for that conic, finding
different points.

So... It took Wison and I a couple of days to find out that this is
an interesting operation. it is easily constructable. The resulting
coordinates are simple. It relates points related by nothing else.
Why has this operation been ignored for the last hundred years?

Friendly from the US,,

Steve

> [WS]
> Can I perhaps add a few more sets of points,
> and fill a few gaps.
>
> Let me write
> sP for the supplement of P,
> aP for the antisupplement of P,
> gP for the isogonal conjugate of P
>
> Function P sP aP

[I have combined my and Wilson's lists using his notation]
>
>> cos A 3, 65, 46
>
> sin A 6 37 9
> csc A 2 42 43
>
> tan A 19 48 610
> cot A 63 31 17

>> sec B 4 73 1745

>
> cotA/2 9 6 1743
> tanA/2 57 55 165

>> sin B/2 266 , --, 164
>> 258 on this line
>> sec B/2 174 --, 503
>> csc B/2 188 --, 361
>> cos B/2 259 ---- 173

>
> These give new and simple forms for the trilinears
> of X610, X1707, X1743, and alternative versions for
> those of X6, X31, X37, X42.
>
> I cannot find much geometry, but do observe that
>
> aP = gP-Ceva conjugate of I
> gP = sp-cross conjugate of I.
>
> Also, (P, sP, aP, I) is harmonic,
> as is (P, aP, sP, X), where
> X is the intersection of IP with the antiorthic axis.

[Non-text portions of this message have been removed]
• ... getting ... As before, we write sP for the supplement of P, aP for the antisupplement of P, gP for the isogonal conjugate of P. ... Not quite, I find that
Message 3 of 21 , Oct 2, 2005
--- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@c...> wrote:
> Following Wison's lead and using his notation, I think I am
getting
> a handle on this "supplement" operation.

As before, we write
sP for the supplement of P,
aP for the antisupplement of P,
gP for the isogonal conjugate of P.

First, some extracts from Steve's message and my comments:

> sP can be constructed as the the cevian quotient gP/Io by connecting
> the ex-incenters to the cevian traces of gP.

Not quite, I find that the cevian quotient gP/I is aP, not sP.

> I am ignoring the cross conjugate operation because it seems too
> complicated for me to understand it as other than a mathematical
> operation. I do notice that it is an involution (to what?) and that
> may make it interesting later.

I try to explain below why the operation is natural and ought to be
included in this theory sooner rather than later.

Also, the operation P -> gaP is not an involution.
We do meet the involution "I-cross conjugate of P" later,
as well as the involution "I-ceva conjugate of P".

> The interesting part is that the isogonals of sP and sgP are
collinear
> to P and gP!

Very nice - I had not noticed this!

> 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.

**********************************************************************
*

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.

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 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)

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.

Regards

Wilson
• I ve been enjoying current mails on supplements etc. Stepping in where even angels would fear to tread I wonder why we can t use cevian product * as we are
Message 4 of 21 , Oct 2, 2005
I've been enjoying current mails on supplements etc. Stepping in where
even angels would fear to tread I wonder why we can't use cevian product *
as we are already using the quotient ­ giviing, I think, sP = g(P*I)

Dick Tahta
• Hi Dick, It s nice to know someone (else) is interested in this topic. As you say, sP = g(P*I), so gsP = P*I = cevapoint of P,I. I think that we were coming to
Message 5 of 21 , Oct 2, 2005
Hi Dick,

It's nice to know someone (else) is interested in this topic.

As you say, sP = g(P*I), so

gsP = P*I = cevapoint of P,I.

I think that we were coming to the idea that sP and gP are
more fundamental, since the other interesting points can be
derived from these two. Also, s,g arise from nne very simple
figure.

But equally we could begin with cevian product and quotient
and isogonal conjugation.

It seems more elegant to have just g,s (and its inverse).

This is probably a matter of taste.

As Steve has observed, s appeared rather early in the study
of the subject.

Regards,

Wilson

--- In Hyacinthos@yahoogroups.com, dick tahta <dick@t...> wrote:
> I've been enjoying current mails on supplements etc. Stepping in
where
> even angels would fear to tread I wonder why we can't use cevian
product *
> as we are already using the quotient ­giving, I think, sP = g(P*I)
>
> Dick Tahta
• Wilson, and now Dick, I first owe a bit of an apology. I have been rereading my old French books and the supplementaire of a point is not :z+x: in
Message 6 of 21 , Oct 2, 2005
Wilson, and now Dick,

I first owe a bit of an apology. I have been rereading my old French
books and the "supplementaire" of a point is not :z+x: in trilinears.
It is that point referred to the incentral triangle.

So I am going to call our operation (which is an entirely valid one)
the supplement, and use the French word for the now no longer in use
idea.

The goal was to produce an incentered analogy to complement. The
problem is that the analogy can be made in two ways. The obvious way
is to consider the equivalent point constructed in the incentral and
excentral triangles. The problem is that, unlike the complement,
there are no nice properties when you do this. These points are not
colinear, nor do they arrange themselves in simple patters. So it is
not obvious what you get from this operation.

But the other choice, the one we are using, enforces a good analogy
to the complement in that the points are collinear and harmonic, but
one has no initial idea of the geometric interpretation of the
supplement and antisupplement. I think we are solving this as we speak.

----

The Jerabek hyperbola has the distinction of having both the isotomic
and isogonic of H, the orthocenter on this hyperbola. We will call
its perspector J.

The supplement has the property that P, gsP, gsgP, gP are collinear.
I call this line the Mineur line, since it is so important in his work.

For H, this is H, 65, 73, O, The isogonal conjugate of this line
line is the Jerabek hyperbols, which contains the conjugates of these
points H, 21, 29, O are on this hyperbola. Since 21 and 29 are weak
and quartile, these points are all on this hyperbola. This in turn
means that the tripolars of H, O, 21, 29 go through the perspector J.
This makes 10 lines that come from this property of the supplement.

The complement has the property that P, tmP, tmtP, tP are collinear
where m is the median (complement) operation. This is the isotomic
version of the Mineur line.

For H this is

H, tO, tK, tH, which means that the isotomic of this line is the
Jerabek hyperbola containing H, tH, O, K, and that the tripolars of
H, tH, O, K go through the perspector. which adds two more lines to
the 10 above.
• Hey Dick and Wison, ... Yea, we seem to have put everyone else to sleep! ... If I remember the ETC glossary correctly, the cevapoint operation is an involution
Message 7 of 21 , Oct 2, 2005
Hey Dick and Wison,

On Oct 2, 2005, at 4:34 PM, Wilson Stothers wrote:

> Hi Dick,
>
> It's nice to know someone (else) is interested in this topic.

Yea, we seem to have put everyone else to sleep!

>
> As you say, sP = g(P*I), so
>
> gsP = P*I = cevapoint of P,I.

If I remember the ETC glossary correctly, the cevapoint operation is
an involution (to what??) so that

gsgsP must then be proportional to P.

---

>
> As Steve has observed, s appeared rather early in the study
> of the subject.
>

I just wrote an apology saying that, although it is true that s ~ :z
+x: was used a hundred years ago, it is not the "supplementaire" that
began this discussion, which is the point constructed in the
incentral triangle. Again, my apologies for that confusion, but this
operation we are talking about may be the more useful version of this
idea.

[Non-text portions of this message have been removed]
• ... I have looked at your translation of Mineur. From this, I think that your apology is not needed. The two notions coincide. If you look at a result in
Message 8 of 21 , Oct 3, 2005
--- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@c...> wrote:
> Wilson, and now Dick,
>
> I first owe a bit of an apology. I have been rereading my old
> French books and the "supplementaire" of a point is not :z+x: in
> trilinears. It is that point referred to the incentral triangle.
>
> So I am going to call our operation (which is an entirely valid one)
> the supplement, and use the French word for the now no longer in use
> idea.
>

I have looked at your translation of Mineur. From this, I think that
your apology is not needed. The two notions coincide.

If you look at a result in section 26 (in your translation):

When a triangle is inscribed in ABC is in perspective to ABC, it is
again in perspective to the antisupplementary triangle, and the second
center of perspective is the antisupplementary to the inverse of the
first.

This translates into more modern language as follows :

The perspector of the anticevian triangle of I (the antisupplementary
triangle) and the cevian triangle of P (= the P-ceva conjugate of I)
is the antisupplementaire of the isogonal conjugate of P,

In our discussion, we found that :

the P-ceva conjugate of I is agP, the antisupplement of the isogonal
conjugate of P.

Thus our antisupplement is the same as Mineur's antisupplementaire.

There is confirmation later (section 31, subsection 2)

certain conics are found to meet at the antisupplementaire of O.
A Cabri sketch confirms that they meet at X(46).
Clark gives this as the H-ceva conjugate of I,
which we interpret as the antisupplement of O = gH.

Note that your translation describes the point as the reflection
of the incentre in the circumcentre, that is X(40), so there is
something amiss here.

Good news?

Wilson

You might like to check Clark's Glossary, the cevapoint
operation is not an involution.
• Wiison, Thanks for all this. This is good news but if the aO = X(46), it is not the circumcenter of the excentral triangle, which is X(40). This means that the
Message 9 of 21 , Oct 3, 2005
Wiison,

Thanks for all this. This is good news but if the aO = X(46), it is
not the circumcenter of the excentral triangle, which is X(40). This
means that the antisupplementaire (I use French for Mineur's version)
has no relation to the excentral triangle, nor I presume does the
supplementaire to the incentral triangle.

My computer results seem to indicate that Io, 3, 46 are colinear with
40 (O of excentral). This line is the Euler line of the excentral
triangle. If so this is interesting and has interesting implications
for the extraversions of these points (these lines must got through O).

So I am glad our version is the same as the old one.

I like the results we have generated, and this operation makes a lot
more sense now.

But I always notice more for aP than for sP.

Steve

On Oct 3, 2005, at 4:29 AM, Wilson Stothers wrote:

> I have looked at your translation of Mineur. From this, I think that
> your apology is not needed. The two notions coincide.
>
> If you look at a result in section 26 (in your translation):
>
> When a triangle is inscribed in ABC is in perspective to ABC, it is
> again in perspective to the antisupplementary triangle, and the second
> center of perspective is the antisupplementary to the inverse of the
> first.
>
> This translates into more modern language as follows :
>
> The perspector of the anticevian triangle of I (the antisupplementary
> triangle) and the cevian triangle of P (= the P-ceva conjugate of I)
> is the antisupplementaire of the isogonal conjugate of P,
>
> In our discussion, we found that :
>
> the P-ceva conjugate of I is agP, the antisupplement of the isogonal
> conjugate of P.
>
> Thus our antisupplement is the same as Mineur's antisupplementaire.
>
> There is confirmation later (section 31, subsection 2)
>
> certain conics are found to meet at the antisupplementaire of O.
> A Cabri sketch confirms that they meet at X(46).
> Clark gives this as the H-ceva conjugate of I,
> which we interpret as the antisupplement of O = gH.
>
> Note that your translation describes the point as the reflection
> of the incentre in the circumcentre, that is X(40), so there is
> something amiss here.
>
> Good news?

[Non-text portions of this message have been removed]
• Wilson, ... 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
Message 10 of 21 , Oct 3, 2005
Wilson,

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