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21975Re: loci related to Taylor circle

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  • rhutson2
    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
      >
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