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Re: Pohoata's concyclic points

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  • Francisco Javier
    Dear Randy, The point at infinity of line O1O2 is X(526). The coordinates of O1 are shown below (T stands for twice the area of the triangle multiplied by
    Message 1 of 3 , Dec 1, 2011
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      Dear Randy,

      The point at infinity of line O1O2 is X(526).

      The coordinates of O1 are shown below (T stands for twice the area of the triangle multiplied by \sqrt{3}).

      The coordinates of O2 are like that of O1, but making the change T -> -T.

      Best regards,

      Francisco Javier.

      ------------------

      Coordinates of O1:

      {a^2 (b - c) (b + c) (a^14 - 8 a^12 b^2 + 27 a^10 b^4 - 50 a^8 b^6 +
      55 a^6 b^8 - 36 a^4 b^10 + 13 a^2 b^12 - 2 b^14 - 8 a^12 c^2 +
      42 a^10 b^2 c^2 - 69 a^8 b^4 c^2 + 25 a^6 b^6 c^2 +
      39 a^4 b^8 c^2 - 39 a^2 b^10 c^2 + 10 b^12 c^2 + 27 a^10 c^4 -
      69 a^8 b^2 c^4 - 3 a^6 b^4 c^4 - 12 a^4 b^6 c^4 +
      39 a^2 b^8 c^4 - 18 b^10 c^4 - 50 a^8 c^6 + 25 a^6 b^2 c^6 -
      12 a^4 b^4 c^6 - 8 a^2 b^6 c^6 + 10 b^8 c^6 + 55 a^6 c^8 +
      39 a^4 b^2 c^8 + 39 a^2 b^4 c^8 + 10 b^6 c^8 - 36 a^4 c^10 -
      39 a^2 b^2 c^10 - 18 b^4 c^10 + 13 a^2 c^12 + 10 b^2 c^12 -
      2 c^14 - 2 a^12 T + 10 a^10 b^2 T - 20 a^8 b^4 T + 20 a^6 b^6 T -
      10 a^4 b^8 T + 2 a^2 b^10 T + 10 a^10 c^2 T - 12 a^8 b^2 c^2 T -
      22 a^6 b^4 c^2 T + 40 a^4 b^6 c^2 T - 16 a^2 b^8 c^2 T -
      20 a^8 c^4 T - 22 a^6 b^2 c^4 T - 38 a^4 b^4 c^4 T +
      22 a^2 b^6 c^4 T - 4 b^8 c^4 T + 20 a^6 c^6 T +
      40 a^4 b^2 c^6 T + 22 a^2 b^4 c^6 T + 8 b^6 c^6 T -
      10 a^4 c^8 T - 16 a^2 b^2 c^8 T - 4 b^4 c^8 T +
      2 a^2 c^10 T), -b^2 (a - c) (a + c) (-2 a^14 + 13 a^12 b^2 -
      36 a^10 b^4 + 55 a^8 b^6 - 50 a^6 b^8 + 27 a^4 b^10 - 8 a^2 b^12 +
      b^14 + 10 a^12 c^2 - 39 a^10 b^2 c^2 + 39 a^8 b^4 c^2 +
      25 a^6 b^6 c^2 - 69 a^4 b^8 c^2 + 42 a^2 b^10 c^2 - 8 b^12 c^2 -
      18 a^10 c^4 + 39 a^8 b^2 c^4 - 12 a^6 b^4 c^4 - 3 a^4 b^6 c^4 -
      69 a^2 b^8 c^4 + 27 b^10 c^4 + 10 a^8 c^6 - 8 a^6 b^2 c^6 -
      12 a^4 b^4 c^6 + 25 a^2 b^6 c^6 - 50 b^8 c^6 + 10 a^6 c^8 +
      39 a^4 b^2 c^8 + 39 a^2 b^4 c^8 + 55 b^6 c^8 - 18 a^4 c^10 -
      39 a^2 b^2 c^10 - 36 b^4 c^10 + 10 a^2 c^12 + 13 b^2 c^12 -
      2 c^14 + 2 a^10 b^2 T - 10 a^8 b^4 T + 20 a^6 b^6 T -
      20 a^4 b^8 T + 10 a^2 b^10 T - 2 b^12 T - 16 a^8 b^2 c^2 T +
      40 a^6 b^4 c^2 T - 22 a^4 b^6 c^2 T - 12 a^2 b^8 c^2 T +
      10 b^10 c^2 T - 4 a^8 c^4 T + 22 a^6 b^2 c^4 T -
      38 a^4 b^4 c^4 T - 22 a^2 b^6 c^4 T - 20 b^8 c^4 T +
      8 a^6 c^6 T + 22 a^4 b^2 c^6 T + 40 a^2 b^4 c^6 T +
      20 b^6 c^6 T - 4 a^4 c^8 T - 16 a^2 b^2 c^8 T - 10 b^4 c^8 T +
      2 b^2 c^10 T), -(a - b) (a + b) c^2 (2 a^14 - 10 a^12 b^2 +
      18 a^10 b^4 - 10 a^8 b^6 - 10 a^6 b^8 + 18 a^4 b^10 -
      10 a^2 b^12 + 2 b^14 - 13 a^12 c^2 + 39 a^10 b^2 c^2 -
      39 a^8 b^4 c^2 + 8 a^6 b^6 c^2 - 39 a^4 b^8 c^2 +
      39 a^2 b^10 c^2 - 13 b^12 c^2 + 36 a^10 c^4 - 39 a^8 b^2 c^4 +
      12 a^6 b^4 c^4 + 12 a^4 b^6 c^4 - 39 a^2 b^8 c^4 + 36 b^10 c^4 -
      55 a^8 c^6 - 25 a^6 b^2 c^6 + 3 a^4 b^4 c^6 - 25 a^2 b^6 c^6 -
      55 b^8 c^6 + 50 a^6 c^8 + 69 a^4 b^2 c^8 + 69 a^2 b^4 c^8 +
      50 b^6 c^8 - 27 a^4 c^10 - 42 a^2 b^2 c^10 - 27 b^4 c^10 +
      8 a^2 c^12 + 8 b^2 c^12 - c^14 + 4 a^8 b^4 T - 8 a^6 b^6 T +
      4 a^4 b^8 T - 2 a^10 c^2 T + 16 a^8 b^2 c^2 T -
      22 a^6 b^4 c^2 T - 22 a^4 b^6 c^2 T + 16 a^2 b^8 c^2 T -
      2 b^10 c^2 T + 10 a^8 c^4 T - 40 a^6 b^2 c^4 T +
      38 a^4 b^4 c^4 T - 40 a^2 b^6 c^4 T + 10 b^8 c^4 T -
      20 a^6 c^6 T + 22 a^4 b^2 c^6 T + 22 a^2 b^4 c^6 T -
      20 b^6 c^6 T + 20 a^4 c^8 T + 12 a^2 b^2 c^8 T + 20 b^4 c^8 T -
      10 a^2 c^10 T - 10 b^2 c^10 T + 2 c^12 T)}

      --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
      >
      > Dear Hyacinthists,
      >
      > The entry in ETC for X(399) = PARRY REFLECTION POINT includes this note:
      >
      > 'Pohoata reports that the following points are concyclic: X(13), X(16), X(110), X(399), X(1337)[sic, should be X(1338)], as are the points X(14), X(15), X(110), X(399), X(1337).'
      >
      > Has anyone studied the circumcircles of these points?
      >
      > I have the following results and questions:
      >
      > The circle through X(13),X(16),X(110),X(399),X(1338) also includes X(2381) (the other intersection besides X(110) with the circumcircle).
      >
      > The circle through X(14),X(15),X(110),X(399),X(1337) also includes X(2380) (the other intersection besides X(110) with the circumcircle).
      >
      > Let O1 be the circumcenter of X(13)X(16)X(110)X(399)X(1338)X(2381).
      > Let O2 be the circumcenter of X(14)X(15)X(110)X(399)X(1337)X(2380).
      > The centroid of X(3)O1O2 = X(351) = Center of the Parry circle.
      >
      > O1 and O2 are not ETC centers, nor are the radical trace or in/exsimilicenters of the two circles. What are the trilinears?
      >
      > Does the line through the centers O1O2 contain any ETC centers? (It is perpendicular to line 3,74, and intersects it at the radical trace of the two circles.) The trilinear pole of this line is also not an ETC center.
      >
      > Best regards,
      > Randy
      >
    • Barry Wolk
      I was able to replace your polynomial of degree 14 with one of degree 6.   The coordinates of O1 are cyclically symmetric, and the first coordinate is    
      Message 2 of 3 , Dec 3, 2011
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        I was able to replace your polynomial of degree 14 with one of degree 6.
         
        The coordinates of O1 are cyclically symmetric, and the first coordinate is
         
         
        ( 3 ( b^2 + c^2 - a^2 )^3 + 3 b^2 c^2 ( 9 a^2 - 5 b^2 - 5 c^2 ) + 2 T ( ( b^2 + c^2 - a^2 )^2 - b^2 c^2 ) )
         
        This is short enough to be listed in ETC.
        --
        Barry Wolk
         
        [Francisco Javier]
        >Dear Randy,
        >
        >The point at infinity of line O1O2 is X(526).
        >
        >The coordinates of O1 are shown below (T stands for twice the area of the triangle multiplied by \sqrt{3}).
        >
        >The coordinates of O2 are like that of O1, but making the change T -> -T.
        >
        >Best regards,
        >
        >Francisco Javier.
        >
        >------------------
        >
        >Coordinates of O1:
        >
        >{a^2 (b - c) (b + c) (a^14 - 8 a^12 b^2 + 27 a^10 b^4 - 50 a^8 b^6 +
        >    55 a^6 b^8 - 36 a^4 b^10 + 13 a^2 b^12 - 2 b^14 - 8 a^12 c^2 +
        >    42 a^10 b^2 c^2 - 69 a^8 b^4 c^2 + 25 a^6 b^6 c^2 +
        >    39 a^4 b^8 c^2 - 39 a^2 b^10 c^2 + 10 b^12 c^2 + 27 a^10 c^4 -
        >    69 a^8 b^2 c^4 - 3 a^6 b^4 c^4 - 12 a^4 b^6 c^4 +
        >    39 a^2 b^8 c^4 - 18 b^10 c^4 - 50 a^8 c^6 + 25 a^6 b^2 c^6 -
        >    12 a^4 b^4 c^6 - 8 a^2 b^6 c^6 + 10 b^8 c^6 + 55 a^6 c^8 +
        >    39 a^4 b^2 c^8 + 39 a^2 b^4 c^8 + 10 b^6 c^8 - 36 a^4 c^10 -
        >    39 a^2 b^2 c^10 - 18 b^4 c^10 + 13 a^2 c^12 + 10 b^2 c^12 -
        >    2 c^14 - 2 a^12 T + 10 a^10 b^2 T - 20 a^8 b^4 T + 20 a^6 b^6 T -
        >    10 a^4 b^8 T + 2 a^2 b^10 T + 10 a^10 c^2 T - 12 a^8 b^2 c^2 T -
        >    22 a^6 b^4 c^2 T + 40 a^4 b^6 c^2 T - 16 a^2 b^8 c^2 T -
        >    20 a^8 c^4 T - 22 a^6 b^2 c^4 T - 38 a^4 b^4 c^4 T +
        >    22 a^2 b^6 c^4 T - 4 b^8 c^4 T + 20 a^6 c^6 T +
        >    40 a^4 b^2 c^6 T + 22 a^2 b^4 c^6 T + 8 b^6 c^6 T -
        >    10 a^4 c^8 T - 16 a^2 b^2 c^8 T - 4 b^4 c^8 T +
        >    2 a^2 c^10 T),
         
        [other 2 coordinates snipped]
         
        >
        >--- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
        >>
        >> Dear Hyacinthists,
        >>
        >> The entry in ETC for X(399) = PARRY REFLECTION POINT includes this note:
        >>
        >> 'Pohoata reports that the following points are concyclic: X(13), X(16), X(110), X(399), X(1337)[sic, should be X(1338)], as are the points X(14), X(15), X(110), X(399), X(1337).'
        >>
        >> Has anyone studied the circumcircles of these points?
        >>
        >> I have the following results and questions:
        >>
        >> The circle through X(13),X(16),X(110),X(399),X(1338) also includes X(2381) (the other intersection besides X(110) with the circumcircle).
        >>
        >> The circle through X(14),X(15),X(110),X(399),X(1337) also includes X(2380) (the other intersection besides X(110) with the circumcircle).
        >>
        >> Let O1 be the circumcenter of X(13)X(16)X(110)X(399)X(1338)X(2381).
        >> Let O2 be the circumcenter of X(14)X(15)X(110)X(399)X(1337)X(2380).
        >> The centroid of X(3)O1O2 = X(351) = Center of the Parry circle.
        >>
        >> O1 and O2 are not ETC centers, nor are the radical trace or in/exsimilicenters of the two circles.  What are the trilinears?
        >>
        >> Does the line through the centers O1O2 contain any ETC centers?  (It is perpendicular to line 3,74, and intersects it at the radical trace of the two circles.)  The trilinear pole of this line is also not an ETC center.
        >>
        >> Best regards,
        >> Randy
        a^2 ( b^2 - c^2 )
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