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Re: [EMHL] Re: supplementaire

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  • Steve Sigur
    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
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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 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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    • Steve Sigur
      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
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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 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]
      • Wilson Stothers
        ... 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
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          --- 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
        • dick tahta
          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
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            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
          • Wilson Stothers
            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
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              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
            • Steve Sigur
              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
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                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.
              • Steve Sigur
                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
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                  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]
                • Wilson Stothers
                  ... 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
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                    --- 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

                    PS (re your other message)
                    You might like to check Clark's Glossary, the cevapoint
                    operation is not an involution.
                  • Steve Sigur
                    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
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                      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]
                    • Steve Sigur
                      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
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                        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
                        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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