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loci related to Taylor circle

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  • rhutson2
    Dear friends, Let ABC be a triangle, and P a point. Let A B C be the pedal triangle of P. Let Ba, Ca be the orthogonal projections of A onto lines CA, AB,
    Message 1 of 5 , Apr 16, 2013
      Dear friends,

      Let ABC be a triangle, and P a point.
      Let A'B'C' be the pedal triangle of P.
      Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB, resp.
      Define Cb, Ab, Ac, Bc cyclically.

      What is the locus of P such that Ba, Ca, Cb, Ab, Ac, Bc lie on a common conic? The locus would include H, for which the conic is the Taylor circle.

      What is the locus of the centers of the conics for P in the above locus? It would include X(394), the center of the Taylor circle.

      Best regards,
      Randy
    • Bernard Gibert
      Dear Randy, ... a quintic with many simple points but only two (I think) ETC centers : X4, X1498. ... seems very difficult... Best regards Bernard [Non-text
      Message 2 of 5 , Apr 17, 2013
        Dear Randy,

        > [RH] Let ABC be a triangle, and P a point.
        > Let A'B'C' be the pedal triangle of P.
        > Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB, resp.
        > Define Cb, Ab, Ac, Bc cyclically.
        >
        > What is the locus of P such that Ba, Ca, Cb, Ab, Ac, Bc lie on a common conic? The locus would include H, for which the conic is the Taylor circle.

        a quintic with many simple points but only two (I think) ETC centers : X4, X1498.



        > What is the locus of the centers of the conics for P in the above locus? It would include X(394), the center of the Taylor circle.

        seems very difficult...

        Best regards

        Bernard

        [Non-text portions of this message have been removed]
      • yiuatfauedu
        Dear Randy and Bernard, [RH] Let ABC be a triangle, and P a point. Let A B C be the pedal triangle of P. Let Ba, Ca be the orthogonal projections of A onto
        Message 3 of 5 , Apr 17, 2013
          Dear Randy and Bernard,

          [RH] Let ABC be a triangle, and P a point.
          Let A'B'C' be the pedal triangle of P.
          Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB, resp.
          Define Cb, Ab, Ac, Bc cyclically.
          What is the locus of P such that Ba, Ca, Cb, Ab, Ac, Bc lie on a common conic? The locus would include H, for which the conic is the Taylor circle.

          [BG]: a quintic with many simple points but only two (I think) ETC centers : X4, X1498.

          *** X(1498) is the Nagel point of the tangential triangle.
          Ab = (0 : S_{AB}+S_{AC}-S_{BC} : S_{AB}+S_{AC}+S_{BC}) and
          Ac = (0 : S_{AB}+S_{AC}+S_{BC} : S_{AB}+S_{AC}+S_{BC})
          are isotomic points on BC, so are Bc, Ba, and Ca, Cb.
          X(1498) is the unique point with this property. The conic is

          (4/S^2)cyclic sum ((a^4S_{AA})/(S_{AB}+S_AC}-S_{BC}))yz - (x+y+z)^2 = 0,
          concentric (and homothetic) with the circumconic with perspector
          ((a^4S_{AA}/(S_{AB}+S_{AC}-S_{BC}):...:...)
          [with (6-9-13)-search number 5.10435062529...]
          and has center
          (a^4(S_{AAAB}+S_{AAAC}+S_{AABB}-S_{AABC}+S_{AACC}-S_{BBCC}/
          (S_{AB}+S_{AC}-S_{BC}) :...:...)
          with (6-9-13) search number 1.09478783248....

          Best regards
          Sincerely
          Paul
        • rhutson2
          Paul, This is interesting: the perspector you mention with ETC search value 5.10435062529 matches the isogonal conjugate of the polar conjugate of X(1073), and
          Message 4 of 5 , Apr 17, 2013
            Paul,

            This is interesting: the perspector you mention with ETC search value 5.10435062529 matches the isogonal conjugate of the polar conjugate of X(1073), and as such would have trilinears (cos^2 A)/(cos A - cos B cos C) : :.

            What do you mean by the notations S_{AB}, etc.? I assume it has to do with Conway notation. Also, are your coordinates trilinears or barycentrics?

            Randy

            --- In Hyacinthos@yahoogroups.com, "yiuatfauedu" <yiu@...> wrote:
            >
            > Dear Randy and Bernard,
            >
            > [RH] Let ABC be a triangle, and P a point.
            > Let A'B'C' be the pedal triangle of P.
            > Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB, resp.
            > Define Cb, Ab, Ac, Bc cyclically.
            > What is the locus of P such that Ba, Ca, Cb, Ab, Ac, Bc lie on a common conic? The locus would include H, for which the conic is the Taylor circle.
            >
            > [BG]: a quintic with many simple points but only two (I think) ETC centers : X4, X1498.
            >
            > *** X(1498) is the Nagel point of the tangential triangle.
            > Ab = (0 : S_{AB}+S_{AC}-S_{BC} : S_{AB}+S_{AC}+S_{BC}) and
            > Ac = (0 : S_{AB}+S_{AC}+S_{BC} : S_{AB}+S_{AC}+S_{BC})
            > are isotomic points on BC, so are Bc, Ba, and Ca, Cb.
            > X(1498) is the unique point with this property. The conic is
            >
            > (4/S^2)cyclic sum ((a^4S_{AA})/(S_{AB}+S_AC}-S_{BC}))yz - (x+y+z)^2 = 0,
            > concentric (and homothetic) with the circumconic with perspector
            > ((a^4S_{AA}/(S_{AB}+S_{AC}-S_{BC}):...:...)
            > [with (6-9-13)-search number 5.10435062529...]
            > and has center
            > (a^4(S_{AAAB}+S_{AAAC}+S_{AABB}-S_{AABC}+S_{AACC}-S_{BBCC}/
            > (S_{AB}+S_{AC}-S_{BC}) :...:...)
            > with (6-9-13) search number 1.09478783248....
            >
            > Best regards
            > Sincerely
            > Paul
            >
          • Paul Yiu
            Dear Randy, Yes, S_{AB} is shorthand for (S_A)(S_B) etc. I always use barycentric coordinates unless the context clearly favors trilinear coordinates. Let s
            Message 5 of 5 , Apr 17, 2013
              Dear Randy,

              Yes, S_{AB} is shorthand for (S_A)(S_B) etc. I always use barycentric coordinates unless the context clearly favors trilinear coordinates.

              Let's check with the trilinear coordinates you gave. With an extra factor a (in the first component), the barycentric coordinates are

              a cos^2 A/(cos A - cos B cos C) : ... : ...
              = a(S_{AA}/(bc)^2) /(S_A/(bc) - S_{BC}/(a^2bc)) : ... : ...
              = a^3S_{AA}/(a^2bc S_A - bcS_{BC}) : ... : ...
              = a^4S_{AA}/(a^2S_A - S_{BC}) : ... : ...

              Yes, it is the same as the one I gave!

              Best regards
              Sincerely
              Paul
              ________________________________________
              From: Hyacinthos@yahoogroups.com [Hyacinthos@yahoogroups.com] on behalf of rhutson2 [rhutson2@...]
              Sent: Wednesday, April 17, 2013 3:37 PM
              To: Hyacinthos@yahoogroups.com
              Subject: [EMHL] Re: loci related to Taylor circle

              Paul,

              This is interesting: the perspector you mention with ETC search value 5.10435062529 matches the isogonal conjugate of the polar conjugate of X(1073), and as such would have trilinears (cos^2 A)/(cos A - cos B cos C) : :.

              What do you mean by the notations S_{AB}, etc.? I assume it has to do with Conway notation. Also, are your coordinates trilinears or barycentrics?

              Randy

              --- In Hyacinthos@yahoogroups.com, "yiuatfauedu" <yiu@...> wrote:
              >
              > Dear Randy and Bernard,
              >
              > [RH] Let ABC be a triangle, and P a point.
              > Let A'B'C' be the pedal triangle of P.
              > Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB, resp.
              > Define Cb, Ab, Ac, Bc cyclically.
              > What is the locus of P such that Ba, Ca, Cb, Ab, Ac, Bc lie on a common conic? The locus would include H, for which the conic is the Taylor circle.
              >
              > [BG]: a quintic with many simple points but only two (I think) ETC centers : X4, X1498.
              >
              > *** X(1498) is the Nagel point of the tangential triangle.
              > Ab = (0 : S_{AB}+S_{AC}-S_{BC} : S_{AB}+S_{AC}+S_{BC}) and
              > Ac = (0 : S_{AB}+S_{AC}+S_{BC} : S_{AB}+S_{AC}+S_{BC})
              > are isotomic points on BC, so are Bc, Ba, and Ca, Cb.
              > X(1498) is the unique point with this property. The conic is
              >
              > (4/S^2)cyclic sum ((a^4S_{AA})/(S_{AB}+S_AC}-S_{BC}))yz - (x+y+z)^2 = 0,
              > concentric (and homothetic) with the circumconic with perspector
              > ((a^4S_{AA}/(S_{AB}+S_{AC}-S_{BC}):...:...)
              > [with (6-9-13)-search number 5.10435062529...]
              > and has center
              > (a^4(S_{AAAB}+S_{AAAC}+S_{AABB}-S_{AABC}+S_{AACC}-S_{BBCC}/
              > (S_{AB}+S_{AC}-S_{BC}) :...:...)
              > with (6-9-13) search number 1.09478783248....
              >
              > Best regards
              > Sincerely
              > Paul
              >




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