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X(5415) & X(5416) coordinates

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  • César Lozada
    Dear geometers: Here are two coordinates pending in ETC. Maybe this information may be of interest for Professor Kimberling. Regards César Lozada. X(5415) =
    Message 1 of 3 , Apr 6, 2013
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      Dear geometers:


      Here are two coordinates pending in ETC.


      Maybe this information may be of interest for Professor Kimberling.


      Regards


      César Lozada.





      X(5415) = 4th KENMOTU HOMOTHETIC CENTER


      Trilinears (pending)

      The A'B'C' defined at X(5412) is homothetic to the extangents triangle, and
      the center of homothety is X(5415). Also, X(5415) = {X(6), X(55)}-harmonic
      conjugate of X(5416). (Randy Hutson, February 9, 2013)

      The coordinates of X(5415) are:

      Trilinears: a*( (2*R + r ) + (s-a) ) : :

      Trilinears: f(A) : f(B) : f(C) , where f(A)=( (2*R + r + s)/R - 2*sin(A)) *
      sin(A)

      Barycentrics f(A)*sin(A) : f(B)*sin(B) : f(C)*sin(C)



      ETC SEARCH : 1.246310683967065.., 1.64076433091358.., 1.929530398674793..



      X(5415) is a major center.

      X(5415) lies on lines : X(6)-X(31), X(19)-X(5412) , X(40)-X(371)






      X(5416) = 5th KENMOTU HOMOTHETIC CENTER


      Trilinears (pending)

      The A'B'C' defined at X(5413) is homothetic to the extangents triangle, and
      the center of homothety is X(5415). Also, X(5416) = {X(6), X(55)}-harmonic
      conjugate of X(5416). (Randy Hutson, February 9, 2013)

      The coordinates of X(5416) are:

      Trilinears: a*( (2*R + r ) – (s-a) ): :

      Trilinears: f(A) : f(B) : f(C) , where f(A) = ( (2*R + r - s)/R +
      2*sin(A))*sin(A)

      Barycentrics f(A)*sin(A) : f(B)*sin(B) : f(C)*sin(C)



      ETC SEARCH : 0.626431598880468.., 1.27040179200258.., 2.472033272576223..

      X(5416) is a major center.

      X(5416) lies on lines : X(6)-X(31), X(19)-X(5413) , X(40)-X(372)





      [Non-text portions of this message have been removed]
    • Antreas Hatzipolakis
      ... what is the meaning of major center ? aph [Non-text portions of this message have been removed]
      Message 2 of 3 , Apr 7, 2013
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        On Sun, Apr 7, 2013 at 7:00 AM, César Lozada <cesar_e_lozada@...>wrote:

        > **
        >
        >


        >
        > X(5415) is a major center.
        >
        >
        > X(5416) is a major center.
        >

        what is the meaning of "major center"?

        aph


        [Non-text portions of this message have been removed]
      • rhutson2
        Very nice, César! ... appropriate value of k (for the ETC reference triangle, it appears to be 391/40), but I could never find a function for k. I wonder if
        Message 3 of 3 , Apr 7, 2013
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          Very nice, César!

          I had figured the trilinears might be in the form 1 + k sin A + cos A :
          : for X(5415) and 1 - k sin A + cos A : : for X(5416), for some
          appropriate value of k (for the ETC reference triangle, it appears to be
          391/40), but I could never find a function for k. I wonder if there are
          related trilinears in your form for X(2066) and X(5414).

          Antreas, to answer your question (and I had to look it up myself to
          remember), a major center is one whose triangle center function alpha is
          a function of angle A (or side a) alone. Since César's function
          contains R, r and s, which themselves are functions of a, b, and c, I
          wonder if they technically qualify as major centers.

          Best regards,
          Randy


          --- In Hyacinthos@yahoogroups.com, César Lozada <cesar_e_lozada@...>
          wrote:
          >
          >
          > Dear geometers:
          >
          >
          > Here are two coordinates pending in ETC.
          >
          >
          > Maybe this information may be of interest for Professor Kimberling.
          >
          >
          > Regards
          >
          >
          > César Lozada.
          >
          >
          >
          >
          >
          > X(5415) = 4th KENMOTU HOMOTHETIC CENTER
          >
          >
          > Trilinears (pending)
          >
          > The A'B'C' defined at X(5412) is homothetic to the extangents
          triangle, and
          > the center of homothety is X(5415). Also, X(5415) = {X(6),
          X(55)}-harmonic
          > conjugate of X(5416). (Randy Hutson, February 9, 2013)
          >
          > The coordinates of X(5415) are:
          >
          > Trilinears: a*( (2*R + r ) + (s-a) ) : :
          >
          > Trilinears: f(A) : f(B) : f(C) , where f(A)=( (2*R + r + s)/R -
          2*sin(A)) *
          > sin(A)
          >
          > Barycentrics f(A)*sin(A) : f(B)*sin(B) : f(C)*sin(C)
          >
          >
          >
          > ETC SEARCH : 1.246310683967065.., 1.64076433091358..,
          1.929530398674793..
          >
          >
          >
          > X(5415) is a major center.
          >
          > X(5415) lies on lines : X(6)-X(31), X(19)-X(5412) , X(40)-X(371)
          >
          >
          >
          >
          >
          >
          > X(5416) = 5th KENMOTU HOMOTHETIC CENTER
          >
          >
          > Trilinears (pending)
          >
          > The A'B'C' defined at X(5413) is homothetic to the extangents
          triangle, and
          > the center of homothety is X(5415). Also, X(5416) = {X(6),
          X(55)}-harmonic
          > conjugate of X(5416). (Randy Hutson, February 9, 2013)
          >
          > The coordinates of X(5416) are:
          >
          > Trilinears: a*( (2*R + r ) – (s-a) ): :
          >
          > Trilinears: f(A) : f(B) : f(C) , where f(A) = ( (2*R + r - s)/R +
          > 2*sin(A))*sin(A)
          >
          > Barycentrics f(A)*sin(A) : f(B)*sin(B) : f(C)*sin(C)
          >
          >
          >
          > ETC SEARCH : 0.626431598880468.., 1.27040179200258..,
          2.472033272576223..
          >
          > X(5416) is a major center.
          >
          > X(5416) lies on lines : X(6)-X(31), X(19)-X(5413) , X(40)-X(372)
          >
          >
          >
          >
          >
          > [Non-text portions of this message have been removed]
          >
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