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Synthetic/Analytic Geometry

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  • Jeff Brooks
    All, What are the differences between synthetic and analytic geometry? Sincerely, Jeff
    Message 1 of 30 , Feb 22, 2006
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      All,

      What are the differences between synthetic and analytic geometry?

      Sincerely, Jeff
    • Jeff Brooks
      ... I ask this question after reading Geometric Construction of Reciprocal Conjugations by Keith Dean and Floor van Lamoen: see Keith Dean and Floor van
      Message 2 of 30 , Feb 22, 2006
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        --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@...> wrote:
        >
        > All,
        >
        > What are the differences between synthetic and analytic geometry?
        >
        > Sincerely, Jeff
        >

        I ask this question after reading "Geometric Construction of Reciprocal
        Conjugations" by Keith Dean and Floor van Lamoen: see

        Keith Dean and Floor van Lamoen, Geometric Construction of Reciprocal
        Conjugations, pp.115--120.

        http://forumgeom.fau.edu/FG2001volume1/FG200116index.html

        Proposition 1. under 2.1 has a proof but 2.2 construction does not have
        a proof ... The answer seems analytic not synthetic in nature.


        I want to understand this analysis better...

        Friendly, Jeff
      • Francois Rideau
        Dear Jeff I don t know the answer to your question. I think it s only a matter of taste. In bygone days, there was a war between synthetic and analytic
        Message 3 of 30 , Feb 23, 2006
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          Dear Jeff
          I don't know the answer to your question. I think it's only a matter of
          taste. In bygone days, there was a war between synthetic and analytic
          geometers. I think now it's over!
          As for Keith-Floor's paper , I think their construction interesting but too
          intricate.
          To get the map:
          f:(x:y:z) --> (u/x:v/y:w/z)
          I think, it's better to use f = g.s
          where s:(x:y:z) --> (1/x:1/y:1/z) is the isotomic conjugation
          and g:(x:y:z) --> (u.x:v.y:w.z) is the collineation with A, B, C
          (vertices of the reference triengle) as fixed points, sending G(1:1:1) to
          P(u:v:w).
          Of course, we must have a simple construction for g. If you are interested,
          I can send you a Cabri-file of this construction.
          Friendly
          François


          [Non-text portions of this message have been removed]
        • Jeff Brooks
          ... matter of ... analytic ... but too ... (1:1:1) to ... interested, ... Thank you Francois once again, I wish to write A , B and C as Keith and Floor
          Message 4 of 30 , Feb 23, 2006
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            --- In Hyacinthos@yahoogroups.com, "Francois Rideau"
            <francois.rideau@...> wrote:
            >
            > Dear Jeff
            > I don't know the answer to your question. I think it's only a
            matter of
            > taste. In bygone days, there was a war between synthetic and
            analytic
            > geometers. I think now it's over!
            > As for Keith-Floor's paper , I think their construction interesting
            but too
            > intricate.
            > To get the map:
            > f:(x:y:z) --> (u/x:v/y:w/z)
            > I think, it's better to use f = g.s
            > where s:(x:y:z) --> (1/x:1/y:1/z) is the isotomic conjugation
            > and g:(x:y:z) --> (u.x:v.y:w.z) is the collineation with A, B, C
            > (vertices of the reference triengle) as fixed points, sending G
            (1:1:1) to
            > P(u:v:w).
            > Of course, we must have a simple construction for g. If you are
            interested,
            > I can send you a Cabri-file of this construction.
            > Friendly
            > François
            >


            Thank you Francois once again,

            I wish to write A", B" and C" as Keith and Floor describe them in
            some kind of a closed, easily understood and plausible, form. That
            is, I wish to be able to duplicate it using basic math (examples
            would provide a great deal of help.)


            Sincerely, Jeff

            P.S. I don't have Cabri so any examples you provide are all the more
            appreciated.
          • Jeff Brooks
            Stated in another way: I want the coordinates for A and cyclic given arbitrary P and Q. More over, I want a proof of the result. Sincerely, Jeff ...
            Message 5 of 30 , Feb 23, 2006
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              Stated in another way:

              I want the coordinates for A" and cyclic given arbitrary P and Q.
              More over, I want a proof of the result.

              Sincerely, Jeff



              --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@...> wrote:
              >
              > --- In Hyacinthos@yahoogroups.com, "Francois Rideau"
              > <francois.rideau@> wrote:
              > >
              > > Dear Jeff
              > > I don't know the answer to your question. I think it's only a
              > matter of
              > > taste. In bygone days, there was a war between synthetic and
              > analytic
              > > geometers. I think now it's over!
              > > As for Keith-Floor's paper , I think their construction
              interesting
              > but too
              > > intricate.
              > > To get the map:
              > > f:(x:y:z) --> (u/x:v/y:w/z)
              > > I think, it's better to use f = g.s
              > > where s:(x:y:z) --> (1/x:1/y:1/z) is the isotomic conjugation
              > > and g:(x:y:z) --> (u.x:v.y:w.z) is the collineation with A, B,
              C
              > > (vertices of the reference triengle) as fixed points, sending G
              > (1:1:1) to
              > > P(u:v:w).
              > > Of course, we must have a simple construction for g. If you are
              > interested,
              > > I can send you a Cabri-file of this construction.
              > > Friendly
              > > François
              > >
              >
              >
              > Thank you Francois once again,
              >
              > I wish to write A", B" and C" as Keith and Floor describe them in
              > some kind of a closed, easily understood and plausible, form. That
              > is, I wish to be able to duplicate it using basic math (examples
              > would provide a great deal of help.)
              >
              >
              > Sincerely, Jeff
              >
              > P.S. I don't have Cabri so any examples you provide are all the
              more
              > appreciated.
              >
            • Nikolaos Dergiades
              Dear Jeff, Francois is right but I want to add my opinion. I think that in our days synthetic in mathematics is the effort, as in computer scientists, for the
              Message 6 of 30 , Feb 23, 2006
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                Dear Jeff,
                Francois is right but I want
                to add my opinion.

                I think that in our days synthetic in
                mathematics is the effort, as in
                computer scientists, for the use of known
                compound tools, for detail encryption.
                It is called in computer language
                information hiding or data encapsulation.

                Best regards
                Nikos Dergiades


                [FR]
                > > I don't know the answer to your question. I think
                > it's only a
                > matter of
                > > taste.







                ___________________________________________________________
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              • Jeff Brooks
                Dear Nikolaos, I have no idea what you are talking about...Could you please elaborate? Sincerely, Jeff
                Message 7 of 30 , Feb 23, 2006
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                  Dear Nikolaos,

                  I have no idea what you are talking about...Could you please
                  elaborate?

                  Sincerely, Jeff

                  --- In Hyacinthos@yahoogroups.com, Nikolaos Dergiades
                  <ndergiades@...> wrote:
                  >
                  > Dear Jeff,
                  > Francois is right but I want
                  > to add my opinion.
                  >
                  > I think that in our days synthetic in
                  > mathematics is the effort, as in
                  > computer scientists, for the use of known
                  > compound tools, for detail encryption.
                  > It is called in computer language
                  > information hiding or data encapsulation.
                  >
                  > Best regards
                  > Nikos Dergiades
                  >
                  >
                  > [FR]
                  > > > I don't know the answer to your question. I think
                  > > it's only a
                  > > matter of
                  > > > taste.
                  >
                  >
                  >
                  >
                  >
                  >
                  >
                  > ___________________________________________________________
                  > ×ñçóéìïðïéåßôå Yahoo!;
                  > ÂáñåèÞêáôå ôá åíï÷ëçôéêÜ ìçíýìáôá (spam); Ôï Yahoo! Mail
                  > äéáèÝôåé ôçí êáëýôåñç äõíáôÞ ðñïóôáóßá êáôÜ ôùí åíï÷ëçôéêþí
                  > ìçíõìÜôùí http://login.yahoo.com/config/mail?.intl=gr
                  >
                • Jeff Brooks
                  By the way, Francois is wrong, he is ignoring, I think, some very important priciples. Sincerely, Jeff
                  Message 8 of 30 , Feb 23, 2006
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                    By the way,

                    Francois is wrong, he is ignoring, I think, some very important
                    priciples.

                    Sincerely, Jeff


                    --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@...> wrote:
                    >
                    > Dear Nikolaos,
                    >
                    > I have no idea what you are talking about...Could you please
                    > elaborate?
                    >
                    > Sincerely, Jeff
                    >
                    > --- In Hyacinthos@yahoogroups.com, Nikolaos Dergiades
                    > <ndergiades@> wrote:
                    > >
                    > > Dear Jeff,
                    > > Francois is right but I want
                    > > to add my opinion.
                    > >
                    > > I think that in our days synthetic in
                    > > mathematics is the effort, as in
                    > > computer scientists, for the use of known
                    > > compound tools, for detail encryption.
                    > > It is called in computer language
                    > > information hiding or data encapsulation.
                    > >
                    > > Best regards
                    > > Nikos Dergiades
                    > >
                    > >
                    > > [FR]
                    > > > > I don't know the answer to your question. I think
                    > > > it's only a
                    > > > matter of
                    > > > > taste.
                    > >
                    > >
                    > >
                    > >
                    > >
                    > >
                    > >
                    > > ___________________________________________________________
                    > > ×ñçóéìïðïéåßôå Yahoo!;
                    > > ÂáñåèÞêáôå ôá åíï÷ëçôéêÜ ìçíýìáôá (spam); Ôï Yahoo! Mail
                    > > äéáèÝôåé ôçí êáëýôåñç äõíáôÞ ðñïóôáóßá êáôÜ ôùí åíï÷ëçôéêþí
                    > > ìçíõìÜôùí http://login.yahoo.com/config/mail?.intl=gr
                    > >
                    >
                  • Jeff Brooks
                    I mean t principles!
                    Message 9 of 30 , Feb 23, 2006
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                      I mean't "principles!"


                      --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@...> wrote:
                      >
                      > By the way,
                      >
                      > Francois is wrong, he is ignoring, I think, some very important
                      > priciples.
                      >
                      > Sincerely, Jeff
                      >
                      >
                      > --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@> wrote:
                      > >
                      > > Dear Nikolaos,
                      > >
                      > > I have no idea what you are talking about...Could you please
                      > > elaborate?
                      > >
                      > > Sincerely, Jeff
                      > >
                      > > --- In Hyacinthos@yahoogroups.com, Nikolaos Dergiades
                      > > <ndergiades@> wrote:
                      > > >
                      > > > Dear Jeff,
                      > > > Francois is right but I want
                      > > > to add my opinion.
                      > > >
                      > > > I think that in our days synthetic in
                      > > > mathematics is the effort, as in
                      > > > computer scientists, for the use of known
                      > > > compound tools, for detail encryption.
                      > > > It is called in computer language
                      > > > information hiding or data encapsulation.
                      > > >
                      > > > Best regards
                      > > > Nikos Dergiades
                      > > >
                      > > >
                      > > > [FR]
                      > > > > > I don't know the answer to your question. I think
                      > > > > it's only a
                      > > > > matter of
                      > > > > > taste.
                      > > >
                      > > >
                      > > >
                      > > >
                      > > >
                      > > >
                      > > >
                      > > > ___________________________________________________________
                      > > > ×ñçóéìïðïéåßôå Yahoo!;
                      > > > ÂáñåèÞêáôå ôá åíï÷ëçôéêÜ ìçíýìáôá (spam); Ôï Yahoo! Mail
                      > > > äéáèÝôåé ôçí êáëýôåñç äõíáôÞ ðñïóôáóßá êáôÜ ôùí åíï÷ëçôéêþí
                      > > > ìçíõìÜôùí http://login.yahoo.com/config/mail?.intl=gr
                      > > >
                      > >
                      >
                    • Jeff Brooks
                      Good grief! Forget the spelling... let s get on with Triangle Geometry! jeff
                      Message 10 of 30 , Feb 23, 2006
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                        Good grief! Forget the spelling... let's get on with Triangle
                        Geometry!

                        jeff

                        --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@...> wrote:
                        >
                        >
                        > I mean't "principles!"
                        >
                        >
                        > --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@> wrote:
                        > >
                        > > By the way,
                        > >
                        > > Francois is wrong, he is ignoring, I think, some very important
                        > > priciples.
                        > >
                        > > Sincerely, Jeff
                        > >
                        > >
                        > > --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@> wrote:
                        > > >
                        > > > Dear Nikolaos,
                        > > >
                        > > > I have no idea what you are talking about...Could you please
                        > > > elaborate?
                        > > >
                        > > > Sincerely, Jeff
                        > > >
                        > > > --- In Hyacinthos@yahoogroups.com, Nikolaos Dergiades
                        > > > <ndergiades@> wrote:
                        > > > >
                        > > > > Dear Jeff,
                        > > > > Francois is right but I want
                        > > > > to add my opinion.
                        > > > >
                        > > > > I think that in our days synthetic in
                        > > > > mathematics is the effort, as in
                        > > > > computer scientists, for the use of known
                        > > > > compound tools, for detail encryption.
                        > > > > It is called in computer language
                        > > > > information hiding or data encapsulation.
                        > > > >
                        > > > > Best regards
                        > > > > Nikos Dergiades
                        > > > >
                        > > > >
                        > > > > [FR]
                        > > > > > > I don't know the answer to your question. I think
                        > > > > > it's only a
                        > > > > > matter of
                        > > > > > > taste.
                        > > > >
                        > > > >
                        > > > >
                        > > > >
                        > > > >
                        > > > >
                        > > > >
                        > > > > ___________________________________________________________
                        > > > > ×ñçóéìïðïéåßôå Yahoo!;
                        > > > > ÂáñåèÞêáôå ôá åíï÷ëçôéêÜ ìçíýìáôá (spam); Ôï Yahoo! Mail
                        > > > > äéáèÝôåé ôçí êáëýôåñç äõíáôÞ ðñïóôáóßá êáôÜ ôùí åíï÷ëçôéêþí
                        > > > > ìçíõìÜôùí http://login.yahoo.com/config/mail?.intl=gr
                        > > > >
                        > > >
                        > >
                        >
                      • Francois Rideau
                        Dear Jeff Given P(x:y:z), to compute barycentrics of A , B , C is easy but rather tedious and I am too lazy to do it. But I don t understand your question. Do
                        Message 11 of 30 , Feb 23, 2006
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                          Dear Jeff
                          Given P(x:y:z), to compute barycentrics of A", B", C" is easy but rather
                          tedious and I am too lazy to do it.
                          But I don't understand your question.
                          Do you want a detailed computation of these barycentrics or another proof
                          that AP and AA" are isogonal lines?

                          I think Keith-Floor's proof of this fact is not very satisfying for they
                          don't use signed distances (normal coordinates).
                          If you call A1, B1, C1 to be symmetric points of P wrt sides BC, CA, AB,
                          then A"B"C" is the medial triangle of A1B1C1 and lines AA", BB", CC" are
                          respectively the perpendicular bissectors of the sides B1C1, C1A1, A1B1, so
                          it is known they are through Q isogonal conjugate of P wrt ABC.

                          Friendy
                          François


                          [Non-text portions of this message have been removed]
                        • Francois Rideau
                          Dear Nikolaos Your opinion is very interesting. Now we have computers, we can get rid of compasses and rules and we can simulate all geometries on our screens.
                          Message 12 of 30 , Feb 23, 2006
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                            Dear Nikolaos
                            Your opinion is very interesting.
                            Now we have computers, we can get rid of compasses and rules and we can
                            simulate all geometries on our screens.

                            Of course it will depend upon the software.
                            For example you can do affine geometry without compass using only rules,
                            translations, dilations and parallels or projective geometry using only
                            rules or circular geometry using only circles and so one.
                            Friendly
                            François


                            [Non-text portions of this message have been removed]
                          • Moses, Peter J. C.
                            Dear Jeff, [JB] http://forumgeom.fau.edu/FG2001volume1/FG200116index.html Proposition 1. under 2.1 has a proof but 2.2 construction does not have a proof ...
                            Message 13 of 30 , Feb 23, 2006
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                              Dear Jeff,

                              [JB]
                              http://forumgeom.fau.edu/FG2001volume1/FG200116index.html

                              Proposition 1. under 2.1 has a proof but 2.2 construction does not have
                              a proof ... The answer seems analytic not synthetic in nature.

                              [JB]
                              I want the coordinates for A" and cyclic given arbitrary P and Q.
                              More over, I want a proof of the result.

                              In Proposition 1, the point A'' is the reflection of P{p,q,r} in the
                              midpoint of B'C', which ends up as
                              A'' = {b^2 c^2 p + b^2 r SB + c^2 q SC, b^2 r SA, c^2 q SA}

                              A'B'C being the pedal of P

                              Seems that A''B''C'' is perspective to the medial triangle for P on the
                              Stammler hyperbola (= tangential Feuerbach)

                              To the Orthic for P on the Neuberg cubic.

                              Best regards,

                              Peter.
                            • Steve Sigur
                              ... Jeff and François, I too think this war is over and is a matter of taste. Many on Hyacinthos seem to feel that synthetic proofs (which do no emphasize
                              Message 14 of 30 , Feb 23, 2006
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                                On Feb 23, 2006, at 3:40 AM, Francois Rideau wrote:

                                > I don't know the answer to your question. I think it's only a
                                > matter of
                                > taste. In bygone days, there was a war between synthetic and analytic
                                > geometers. I think now it's over!

                                Jeff and François,

                                I too think this war is over and is a matter of taste. Many on
                                Hyacinthos seem to feel that synthetic proofs (which do no emphasize
                                algebra) are superior. But I often work next to a very good geometer
                                whose aims are more classical than most Hyacinthians but who uses
                                algebra almost all the time simply because you can always get an
                                answer with algebra).

                                The reason there is no argument anymore is that coordinates are
                                inherent in synthetic methods, so all one can do is state one's
                                preference). Coxeter's book on the projective plane has a nice
                                discussion of the role and inevitability of coordinatesl

                                I have also been writing on my web site that Euclid inevitably leads
                                to abstract algebra (From Euclid to Abstract Algebra, just revised),
                                which is more than, but includes, coordinates. (Euclid is very
                                interesting to read!).

                                Euclid is really the geometry of points and segments. If you use
                                lines rather than segments, then coordinate methods become very
                                effective and I think it was projective geometry that moved geometers
                                away from synthetic methods.

                                For me, I call myself a non-synthetic geometer because the tasks I
                                set myself are takes far removed from the scope of synthetic
                                geometry, which is good at finding properties of particular geometric
                                objects. I am interested in groups (often very large groups) of
                                objects for which I am finding purely algebraic methods very successful.

                                Jeff, Enjoy your investigations into conjugates!

                                Steve



                                [Non-text portions of this message have been removed]
                              • Steve Sigur
                                I think cabri.com has a reader for both mac and pc for free. Steve ... [Non-text portions of this message have been removed]
                                Message 15 of 30 , Feb 23, 2006
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                                  I think cabri.com has a reader for both mac and pc for free.

                                  Steve


                                  On Feb 23, 2006, at 3:57 AM, Jeff Brooks wrote:

                                  > P.S. I don't have Cabri so any examples you provide are all the more
                                  > appreciated.



                                  [Non-text portions of this message have been removed]
                                • Francois Rideau
                                  Dear Steve You are right. We cannot forget algebra, even if we enjoy synthetic geometry, we know that a good analytic proof may be as fine as a synthetic one.
                                  Message 16 of 30 , Feb 24, 2006
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                                    Dear Steve
                                    You are right. We cannot forget algebra, even if we enjoy synthetic
                                    geometry, we know that a good analytic proof may be as fine as a synthetic
                                    one.
                                    And what is ETC, if not only repeated and tedious computations of
                                    barycentric coordinates?
                                    Besides even simple problems in triangle geometry are better solved by
                                    higher dimensional tools as circulant determinants or some spaces of affine
                                    maps and so one.
                                    They are also other tools in triangle geometry, complex numbers for example
                                    in Morley's books or June Lester's papers.
                                    We know that Morley was very shrewd using complex numbers and I still don't
                                    understand the ideas (as Galois theory) hidden behind some of his
                                    computations. They is still some work to decipher his book!
                                    Friendly
                                    François


                                    [Non-text portions of this message have been removed]
                                  • Wilson Stothers
                                    Dear Jeff, Francois,Steve (and everyone else) We seem to be agreed that syntheic = analytic. As to the details of Floor s paper : Jeff, I do not think it is
                                    Message 17 of 30 , Feb 24, 2006
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                                      Dear Jeff, Francois,Steve (and everyone else)

                                      We seem to be agreed that syntheic = analytic.

                                      As to the details of Floor's paper :

                                      Jeff, I do not think it is worth the effort to work through A",B",C".
                                      Floor uses this result to motivate a construction for the image of a
                                      point Z under the reciprocal conjugation which swaps X and Y.
                                      In itself, it is simply a bad way to find K.

                                      If X = x1:x2:x3, Y = y1:y2:y3, Z = z1:z2:z3, then the image we want
                                      has barycentrics x1y1/z1 and so on.

                                      Floor's construction is not particularly simple, and fails when X = Y.

                                      My way of constructing this image is no simpler, but covers all cases.

                                      Notation :
                                      U.V denotes the line joining points U and V
                                      For a point W = w1:w2:w3
                                      Wa = 0:w2:w3, the intersection of B.C and A.W, and similarly Wb, Wc
                                      aW = 0:w2:-w3, the intersection of B.C and Wb.Wc, and similarly bW, cW

                                      Construction :
                                      cZYa = intersection of C.A and cZ.Ya
                                      bZXa = intersection of B.A and bZ.Xa
                                      Abc = intersection of B.cZYa and C.bZXa

                                      Then define Bca, Cab similarly

                                      The lines A.Abc, B.Bca, C.Cab meet in a point
                                      This has coordinates x1y1/z1:x2y2/z2:x3y3/z3
                                      Obviously you need only construct two of the lines.

                                      The barycentric calculations are very easy.
                                      It is enough to check that Abc has the form *:x2y2/z2:x3y3/z3,
                                      where * is irrelevant.

                                      This gives the reciprocal conjugate, as intended. Also

                                      With Z = G, it gives the barycentric product of X and Y
                                      With Y = G, it gives the barycentric quotient of X by Z

                                      Note: When Z = G, the points aZ, bZ, cZ are infinite.
                                      To construct, say, the line aZ.W, we simply draw the line
                                      at W which is parallel to BC.

                                      Regards

                                      Wilson





                                      --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@...> wrote:
                                      >
                                      > --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@> wrote:
                                      > >
                                      > > All,
                                      > >
                                      > > What are the differences between synthetic and analytic geometry?
                                      > >
                                      > > Sincerely, Jeff
                                      > >
                                      >
                                      > I ask this question after reading "Geometric Construction of
                                      Reciprocal
                                      > Conjugations" by Keith Dean and Floor van Lamoen: see
                                      >
                                      > Keith Dean and Floor van Lamoen, Geometric Construction of
                                      Reciprocal
                                      > Conjugations, pp.115--120.
                                      >
                                      > http://forumgeom.fau.edu/FG2001volume1/FG200116index.html
                                      >
                                      > Proposition 1. under 2.1 has a proof but 2.2 construction does not
                                      have
                                      > a proof ... The answer seems analytic not synthetic in nature.
                                      >
                                      >
                                      > I want to understand this analysis better...
                                      >
                                      > Friendly, Jeff
                                      >
                                    • Jeff Brooks
                                      Yes, I cannot see the forest for the trees. I wish to see someone overcome the tedious nature of this type of problem and develop a solution. Sincerely, Jeff
                                      Message 18 of 30 , Feb 24, 2006
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                                        Yes,

                                        I cannot see the forest for the trees. I wish to see someone
                                        overcome the tedious nature of this type of problem and develop a
                                        solution.

                                        Sincerely, Jeff


                                        --- In Hyacinthos@yahoogroups.com, "Francois Rideau"
                                        <francois.rideau@...> wrote:
                                        >
                                        > Dear Jeff
                                        > Given P(x:y:z), to compute barycentrics of A", B", C" is easy but
                                        rather
                                        > tedious and I am too lazy to do it.
                                        > But I don't understand your question.
                                        > Do you want a detailed computation of these barycentrics or another
                                        proof
                                        > that AP and AA" are isogonal lines?
                                        >
                                        > I think Keith-Floor's proof of this fact is not very satisfying for
                                        they
                                        > don't use signed distances (normal coordinates).
                                        > If you call A1, B1, C1 to be symmetric points of P wrt sides BC,
                                        CA, AB,
                                        > then A"B"C" is the medial triangle of A1B1C1 and lines AA", BB",
                                        CC" are
                                        > respectively the perpendicular bissectors of the sides B1C1, C1A1,
                                        A1B1, so
                                        > it is known they are through Q isogonal conjugate of P wrt ABC.
                                        >
                                        > Friendy
                                        > François
                                        >
                                        >
                                        > [Non-text portions of this message have been removed]
                                        >
                                      • Jeff Brooks
                                        Dear Wilson, I am not in agreement here. If I understand the definitions correctly, it s much easier to move from given synthetic constructions to unknown
                                        Message 19 of 30 , Feb 28, 2006
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                                          Dear Wilson,

                                          I am not in agreement here. If I understand the definitions correctly,
                                          it's much easier to move from given synthetic constructions to unknown
                                          analytic ones.

                                          The development of synthetic methods seems more difficult.

                                          Sincerely, Jeff

                                          P.S. I'm still working all the wonderful information I received.



                                          --- In Hyacinthos@yahoogroups.com, "Wilson Stothers" <wws@...> wrote:
                                          >
                                          > Dear Jeff, Francois,Steve (and everyone else)
                                          >
                                          > We seem to be agreed that syntheic = analytic.
                                          >
                                        • Jeff Brooks
                                          There is nothing in my repertoire to suggest that analytic = synthetic. Jeff ... correctly, ... unknown
                                          Message 20 of 30 , Feb 28, 2006
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                                            There is nothing in my repertoire to suggest that analytic =
                                            synthetic.

                                            Jeff


                                            --- In Hyacinthos@yahoogroups.com, "Jeff Brooks" <trigeom@...> wrote:
                                            >
                                            > Dear Wilson,
                                            >
                                            > I am not in agreement here. If I understand the definitions
                                            correctly,
                                            > it's much easier to move from given synthetic constructions to
                                            unknown
                                            > analytic ones.
                                            >
                                            > The development of synthetic methods seems more difficult.
                                            >
                                            > Sincerely, Jeff
                                            >
                                            > P.S. I'm still working all the wonderful information I received.
                                            >
                                            >
                                            >
                                            > --- In Hyacinthos@yahoogroups.com, "Wilson Stothers" <wws@> wrote:
                                            > >
                                            > > Dear Jeff, Francois,Steve (and everyone else)
                                            > >
                                            > > We seem to be agreed that syntheic = analytic.
                                            > >
                                            >
                                          • Steve Sigur
                                            ... I had a long discussion about this with Conway a few years ago. He uses coordinate methods almost all the time, but of course he had to invent many of
                                            Message 21 of 30 , Mar 1 6:14 AM
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                                              On Feb 28, 2006, at 10:48 PM, Jeff Brooks wrote:

                                              > The development of synthetic methods seems more difficult.



                                              I had a long discussion about this with Conway a few years ago. He
                                              uses coordinate methods almost all the time, but of course he had to
                                              invent many of those methods to make them effective. I have often
                                              thought that he wills the mathematics to work simply because he
                                              believes so strongly that it will work.

                                              I asked him why he did not do more proofs in the old geometric style.
                                              He said that each synthetic proof was unique, that the path from
                                              givens to conclusion was different for every proof and consequently
                                              often difficult to find, and not very generalizable when you do. He
                                              said that coordinates always work, although sometimes they are not
                                              pretty.

                                              I thought "yes, they always work when you are John Conway."

                                              I then remarked that I thought that one leans more from algebra
                                              because, just be being algebra, it often applies for a whole class of
                                              things.

                                              A great example of this last is simply to realize that if you write
                                              ETC in barycentrics, all the affine information of the plane is
                                              contained there in. Just replace a,b,c with l, m, n.

                                              Steve




                                              [Non-text portions of this message have been removed]
                                            • Jeff Brooks
                                              ... to ... style. ... consequently ... He ... of ... write ... hmmn well, there is certainly a lot to be said about willpower. Jeff
                                              Message 22 of 30 , Mar 1 7:30 AM
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                                                --- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@...> wrote:
                                                >
                                                >
                                                > On Feb 28, 2006, at 10:48 PM, Jeff Brooks wrote:
                                                >
                                                > > The development of synthetic methods seems more difficult.
                                                >
                                                >
                                                >
                                                > I had a long discussion about this with Conway a few years ago. He
                                                > uses coordinate methods almost all the time, but of course he had
                                                to
                                                > invent many of those methods to make them effective. I have often
                                                > thought that he wills the mathematics to work simply because he
                                                > believes so strongly that it will work.
                                                >
                                                > I asked him why he did not do more proofs in the old geometric
                                                style.
                                                > He said that each synthetic proof was unique, that the path from
                                                > givens to conclusion was different for every proof and
                                                consequently
                                                > often difficult to find, and not very generalizable when you do.
                                                He
                                                > said that coordinates always work, although sometimes they are not
                                                > pretty.
                                                >
                                                > I thought "yes, they always work when you are John Conway."
                                                >
                                                > I then remarked that I thought that one leans more from algebra
                                                > because, just be being algebra, it often applies for a whole class
                                                of
                                                > things.
                                                >
                                                > A great example of this last is simply to realize that if you
                                                write
                                                > ETC in barycentrics, all the affine information of the plane is
                                                > contained there in. Just replace a,b,c with l, m, n.
                                                >
                                                > Steve
                                                >


                                                hmmn well, there is certainly a lot to be said about willpower.

                                                Jeff
                                              • Ma. de la Paz Alvarez
                                                Hola, (hi) to all of you! About synthethic/ analityc methods I just wish to add a teacher´s point of view: I ve been teaching the Modern Geometry course for a
                                                Message 23 of 30 , Mar 1 10:45 AM
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                                                  Hola, (hi) to all of you!

                                                  About synthethic/ analityc methods I just wish to add a teacher´s point of view: I've been teaching the Modern Geometry course for a loooooong time, and as result of the lack of interest in geometry in the pre university years curricula, students can "rely" on analityc methods without any idea of what is going on geometrically. Once they get the geometric flavour even analytic methods work better.

                                                  As fot the synthetic demonstrations students tend to look at them as "magic" without any method....but eventually they can actually work on demonstrations of their own.

                                                  Now, don't even think this happens with most of the students: as a matter of fact, this course has the greatest reprobation average in our Science School over any course in physics, biology and mathematics.

                                                  On the second course , which is optative and is mainly over modern geometry of the circle (Shively and Altshiller's books), we even have been able to work on hyperbolic synthetic geometry thus preparing students for complex variable methods.

                                                  So yes, I see (I live actually) the difference between these two methods.

                                                  Just to finish a quote from three students of this semester which are taking at the same time Modern Geometry II and Analytic Geometry II: talking about a problem in the analytic geometry course they said:
                                                  "it's so simple to see it synthetically and so tiresome to work it analytically"

                                                  Best wishes
                                                  María de la Paz
                                                  PS I hope my spelling was not too bad......sorry!


                                                  ---------------------------------
                                                  Do You Yahoo!? La mejor conexión a Internet y 2GB extra a tu correo por $100 al mes. http://net.yahoo.com.mx

                                                  [Non-text portions of this message have been removed]
                                                • Jeff Brooks
                                                  ... point of view: I ve been teaching the Modern Geometry course for a loooooong time, and as result of the lack of interest in geometry in the pre university
                                                  Message 24 of 30 , Mar 2 9:06 AM
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                                                    --- In Hyacinthos@yahoogroups.com, "Ma. de la Paz Alvarez"
                                                    <madelapaix@...> wrote:
                                                    >
                                                    > Hola, (hi) to all of you!
                                                    >
                                                    > About synthethic/ analityc methods I just wish to add a teacher´s
                                                    point of view: I've been teaching the Modern Geometry course for a
                                                    loooooong time, and as result of the lack of interest in geometry in
                                                    the pre university years curricula, students can "rely" on analityc
                                                    methods without any idea of what is going on geometrically. Once they
                                                    get the geometric flavour even analytic methods work better.
                                                    >
                                                    > As fot the synthetic demonstrations students tend to look at them
                                                    as "magic" without any method....but eventually they can actually
                                                    work on demonstrations of their own.
                                                    >
                                                    > Now, don't even think this happens with most of the students: as a
                                                    matter of fact, this course has the greatest reprobation average in
                                                    our Science School over any course in physics, biology and
                                                    mathematics.
                                                    >
                                                    > On the second course , which is optative and is mainly over modern
                                                    geometry of the circle (Shively and Altshiller's books), we even have
                                                    been able to work on hyperbolic synthetic geometry thus preparing
                                                    students for complex variable methods.
                                                    >
                                                    > So yes, I see (I live actually) the difference between these two
                                                    methods.
                                                    >
                                                    > Just to finish a quote from three students of this semester which
                                                    are taking at the same time Modern Geometry II and Analytic Geometry
                                                    II: talking about a problem in the analytic geometry course they said:
                                                    > "it's so simple to see it synthetically and so tiresome to work it
                                                    analytically"
                                                    >
                                                    > Best wishes
                                                    > María de la Paz
                                                    > PS I hope my spelling was not too bad......sorry!
                                                    >
                                                    >

                                                    Dear María,

                                                    Thank you very much for these comments; I am definitely in agreement
                                                    with your student's quote! At times I am even more frustrated, as
                                                    I'm sure others might be, with the enormous amount of information
                                                    available to us in this digital world. With computers, we can obtain
                                                    tons of data but I tend to get lost in trying to decipher meaning
                                                    from it and neither synthetic nor analytic methods alone seem to
                                                    suffice. Seems the best is to work from both sides toward a common
                                                    middle ground where hopefully better understanding is found.

                                                    Sincerely,
                                                    Jeff Brooks
                                                    Tulsa, OK
                                                  • Jeff Brooks
                                                    Dear François, Is Keith-Floor s paper really so intricate after all? Jeff ... matter of ... analytic ... but too ... (1:1:1) to ... interested,
                                                    Message 25 of 30 , Mar 9 10:06 PM
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                                                      Dear François,

                                                      Is Keith-Floor's paper really so intricate after all?

                                                      Jeff


                                                      >
                                                      > Dear Jeff
                                                      > I don't know the answer to your question. I think it's only a
                                                      matter of
                                                      > taste. In bygone days, there was a war between synthetic and
                                                      analytic
                                                      > geometers. I think now it's over!
                                                      > As for Keith-Floor's paper , I think their construction interesting
                                                      but too
                                                      > intricate.
                                                      > To get the map:
                                                      > f:(x:y:z) --> (u/x:v/y:w/z)
                                                      > I think, it's better to use f = g.s
                                                      > where s:(x:y:z) --> (1/x:1/y:1/z) is the isotomic conjugation
                                                      > and g:(x:y:z) --> (u.x:v.y:w.z) is the collineation with A, B, C
                                                      > (vertices of the reference triengle) as fixed points, sending G
                                                      (1:1:1) to
                                                      > P(u:v:w).
                                                      > Of course, we must have a simple construction for g. If you are
                                                      interested,
                                                      > I can send you a Cabri-file of this construction.
                                                      > Friendly
                                                      > François
                                                      >
                                                      >
                                                      > [Non-text portions of this message have been removed]
                                                      >
                                                    • Francois Rideau
                                                      Dear Jeff Yes I read this paper and as I have said it, I am not quite satisfied with some proofs. I have to draw many Cabri pictures of myself to understand
                                                      Message 26 of 30 , Mar 10 1:58 AM
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                                                        Dear Jeff
                                                        Yes I read this paper and as I have said it, I am not quite satisfied with
                                                        some proofs.
                                                        I have to draw many Cabri pictures of myself to understand what they were
                                                        talking about.
                                                        I hope now you are satisfied with my explanation of the chuff-chuffs?
                                                        Friendly
                                                        François
                                                        PS
                                                        Fred Lang send me a Cuppens's Cabri macro drawing cubic knowing 9 points and
                                                        it works properly!
                                                        In particular, I was able to draw the Neuberg cubic very easily!
                                                        I only have some difficulty to choose 9 points among all knwn ETC on it!
                                                        Thanks Fred!


                                                        [Non-text portions of this message have been removed]
                                                      • Jeff Brooks
                                                        Thank you François for answering my posts. Thank you more for allowing me to answer in the order I choose. This one is easier! ... satisfied with ...
                                                        Message 27 of 30 , Mar 10 2:40 AM
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                                                          Thank you François for answering my posts. Thank you more for
                                                          allowing me to answer in the order I choose. This one is easier!

                                                          >
                                                          > Dear Jeff
                                                          > Yes I read this paper and as I have said it, I am not quite
                                                          satisfied with
                                                          > some proofs.

                                                          Neither am I!

                                                          > I have to draw many Cabri pictures of myself to understand what
                                                          they were
                                                          > talking about.
                                                          > I hope now you are satisfied with my explanation of the chuff-
                                                          chuffs?

                                                          I'll get there eventually...

                                                          > Friendly
                                                          > François
                                                          > PS
                                                          > Fred Lang send me a Cuppens's Cabri macro drawing cubic knowing 9
                                                          points and
                                                          > it works properly!
                                                          > In particular, I was able to draw the Neuberg cubic very easily!
                                                          > I only have some difficulty to choose 9 points among all knwn ETC
                                                          on it!
                                                          > Thanks Fred!
                                                          >
                                                          >
                                                          > [Non-text portions of this message have been removed]
                                                          >
                                                        • Wilson Stothers
                                                          Dear Francois et al, Could you or Fred possibly put the macro in the file area? Come to think of it, it might be nice to have an archive of Cabri macros for
                                                          Message 28 of 30 , Mar 10 7:06 AM
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                                                            Dear Francois et al,

                                                            Could you or Fred possibly put the macro in the file area?

                                                            Come to think of it, it might be nice to have an archive of
                                                            Cabri macros for the group. Any thoughts

                                                            Wilson

                                                            --- In Hyacinthos@yahoogroups.com, "Francois Rideau"
                                                            <francois.rideau@...> wrote:

                                                            > Fred Lang send me a Cuppens's Cabri macro drawing cubic knowing 9
                                                            points and
                                                            > it works properly!
                                                            > In particular, I was able to draw the Neuberg cubic very easily!
                                                            > I only have some difficulty to choose 9 points among all knwn ETC
                                                            on it!
                                                            > Thanks Fred!
                                                            >
                                                            >
                                                          • fredlangch
                                                            Dear hyacinthers I finally success to join Mr Roger Cuppens and he agree i can put its 235 macros on conics and cubics for Cabri in the group. Does Antreas
                                                            Message 29 of 30 , Mar 23 8:43 AM
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                                                              Dear hyacinthers
                                                              I finally success to join Mr Roger Cuppens and he agree
                                                              i can put its 235 macros on conics and cubics for Cabri in the group.

                                                              Does Antreas agree ?

                                                              If yes, how can I send it?

                                                              Regards.

                                                              Fred


                                                              --- In Hyacinthos@yahoogroups.com, "Wilson Stothers" <wws@...> wrote:
                                                              >
                                                              > Dear Francois et al,
                                                              >
                                                              > Could you or Fred possibly put the macro in the file area?
                                                              >
                                                              > Come to think of it, it might be nice to have an archive of
                                                              > Cabri macros for the group. Any thoughts
                                                              >
                                                              > Wilson
                                                              >
                                                              > --- In Hyacinthos@yahoogroups.com, "Francois Rideau"
                                                              > <francois.rideau@> wrote:
                                                              >
                                                              > > Fred Lang send me a Cuppens's Cabri macro drawing cubic knowing 9
                                                              > points and
                                                              > > it works properly!
                                                              > > In particular, I was able to draw the Neuberg cubic very easily!
                                                              > > I only have some difficulty to choose 9 points among all knwn ETC
                                                              > on it!
                                                              > > Thanks Fred!
                                                              > >
                                                              > >
                                                              >
                                                            • Jeff
                                                              Dear Nikos and Francois, This is an update to an old post: I think I just now found what you guys were describing: Here, Ian Stewart writes Bourbakism was an
                                                              Message 30 of 30 , Sep 2, 2007
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                                                                Dear Nikos and Francois,

                                                                This is an update to an old post:

                                                                I think I just now found what you guys were describing:

                                                                Here, Ian Stewart writes "Bourbakism was an example of what computer
                                                                scientists now call data compression."

                                                                http://links.jstor.org/sici?sici=0025-5572%28199511%292%3A79%3A486%
                                                                3C496%3ABBPSIM%3E2.0.CO%3B2-E&size=LARGE&origin=JSTOR-enlargePage

                                                                OFF SUBJECT:
                                                                Antreas, I posted another message I think on Friday still pending
                                                                approval -- Could you please delete it -- Thank you,

                                                                Sincerely, Jeff


                                                                > Dear Nikolaos,
                                                                >
                                                                > I have no idea what you are talking about...Could you please
                                                                > elaborate?
                                                                >
                                                                > Sincerely, Jeff
                                                                >
                                                                > --- In Hyacinthos@yahoogroups.com, Nikolaos Dergiades
                                                                > <ndergiades@> wrote:
                                                                > >
                                                                > > Dear Jeff,
                                                                > > Francois is right but I want
                                                                > > to add my opinion.
                                                                > >
                                                                > > I think that in our days synthetic in
                                                                > > mathematics is the effort, as in
                                                                > > computer scientists, for the use of known
                                                                > > compound tools, for detail encryption.
                                                                > > It is called in computer language
                                                                > > information hiding or data encapsulation.
                                                                > >
                                                                > > Best regards
                                                                > > Nikos Dergiades
                                                                > >
                                                                > >
                                                                > > [FR]
                                                                > > > > I don't know the answer to your question. I think
                                                                > > > > it's only a matter of taste.
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