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21973Re: [EMHL] loci related to Taylor circle

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  • yiuatfauedu
    Apr 17, 2013
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      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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