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X(5394) CONGRUENT INCIRCLES POINT seems not to be unique

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  • César Lozada
    Dear geometers: I have found numerically at least other three points determining congruent incircles as in X(5394). In the SEARCH triangle (6,9,13): Pa:
    Message 1 of 2 , Jan 12, 2013
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      Dear geometers:



      I have found numerically at least other three points determining congruent
      incircles as in X(5394).



      In the SEARCH triangle (6,9,13):



      Pa: (u,v,w)=( -2.5886025937.., 3.4499648937.., 2.4469669068..) ,
      inradius=1.062515160982..

      Pb: (u,v,w)=( 6.009055300265.., -2.894577976679.., 2.871192942765..),
      inradius=1.247930440435..

      Pc: (u,v,w)=) 9.637881480452.., 6.105715434464.., -5.034622271032..),
      inradius=2.194838933700..



      Note that each of these points is outside the triangle.



      The given X(5394) has (u,v,w)=( 1.791642688432.., 1.705729272515..,
      1.632862975866..), inradius=0.803384896397… This point is interior to ABC.
      IMHO, X(5394) should be specified that it is an interior point of the
      triangle.



      I’d like to check the work by Noam Elkies in Mathematics Magazine 60 (1987)
      117, cited by ETC in X(5394) and proving the existence of this point.
      Anybody can share it with me?

      Thanks in advance.



      César Lozada











      [Non-text portions of this message have been removed]
    • Randy Hutson
      Dear César, Could Pa, Pb, Pc be extraversions of X(5394)?  I also wondered if PaPbPc might be perspective to ABC, but this turns out not to be the case.
      Message 2 of 2 , Jan 14, 2013
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        Dear César,

        Could Pa, Pb, Pc be extraversions of X(5394)?  I also wondered if PaPbPc might be perspective to ABC, but this turns out not to be the case.


        Randy




        >________________________________
        > From: César Lozada <cesar_e_lozada@...>
        >To: Hyacinthos@yahoogroups.com
        >Sent: Saturday, January 12, 2013 8:24 PM
        >Subject: [EMHL]4X(539) CONGRUENT INCIRCLES POINT seems not to be unique
        >
        >

        >Dear geometers:
        >
        >I have found numerically at least other three points determining congruent
        >incircles as in X(5394).
        >
        >In the SEARCH triangle (6,9,13):
        >
        >Pa: (u,v,w)=( -2.5886025937.., 3.4499648937.., 2.4469669068..) ,
        >inradius=1.062515160982..
        >
        >Pb: (u,v,w)=( 6.009055300265.., -2.894577976679.., 2.871192942765..),
        >inradius=1.247930440435..
        >
        >Pc: (u,v,w)=) 9.637881480452.., 6.105715434464.., -5.034622271032..),
        >inradius=2.194838933700..
        >
        >Note that each of these points is outside the triangle.
        >
        >The given X(5394) has (u,v,w)=( 1.791642688432.., 1.705729272515..,
        >1.632862975866..), inradius=0.803384896397… This point is interior to ABC.
        >IMHO, X(5394) should be specified that it is an interior point of the
        >triangle.
        >
        >I’d like to check the work by Noam Elkies in Mathematics Magazine 60 (1987)
        >117, cited by ETC in X(5394) and proving the existence of this point.
        >Anybody can share it with me?
        >
        >Thanks in advance.
        >
        >César Lozada
        >
        >[Non-text portions of this message have been removed]
        >
        >
        >
        >
        >

        [Non-text portions of this message have been removed]
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