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Re: A lonely and isolated center

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  • Antreas
    ... This is true for any point P instead of G (Paul Yiu) And also for cevians of symmedian point instead of the cevians of Incenter. Simple applications of
    Message 1 of 3 , Apr 7, 2013
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      --- In Hyacinthos@yahoogroups.com, C├ęsar Lozada <cesar_e_lozada@...> wrote:
      >
      > Let A'B'C' be the medial triangle of ABC and G its centroid.
      >
      > Let A" be the intersection of the BC with the angle bisector of B'GC'. Build
      > B" and C" cyclically.
      >
      >
      >
      > The triangles A" B" C" and ABC are perspective with perspector X = (
      > a*sqrt( -a^2 + 2*b^2+2*c^2) : : )

      This is true for any point P instead of G
      (Paul Yiu)

      And also for cevians of symmedian point instead of the cevians
      of Incenter.

      Simple applications of Ceva theorem.

      APH
    • yiuatfauedu
      Dear Cesar and Antreas, [CL]: Let A B C be the medial triangle of ABC and G its centroid. Let A be the intersection of the BC with the angle bisector of
      Message 2 of 3 , Apr 8, 2013
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        Dear Cesar and Antreas,

        [CL]: Let A'B'C' be the medial triangle of ABC and G its centroid.
        Let A" be the intersection of the BC with the angle bisector of B'GC'. Build B" and C" cyclically.
        The triangles A" B" C" and ABC are perspective with perspector
        X = (a*sqrt( -a^2 + 2*b^2+2*c^2) : : )
        ...

        [APH]: This is true for any point P instead of G.

        *** Yes, Antreas. But Cesar, I do not get your coordinates. In homogeneous barycentric coordinates, the perspector is
        (1/m_a : 1/m_b : 1/m_c) where m_a etc are the lengths of the medians. With m_a = sqrt(2b^2+2c^2-a^2)/2 etc, this is
        (1/sqrt{2b^2+2c^2-a^2) : ... : ...).
        The trilinear coordinates are
        (1/(a*sqrt(2b^2+2c^2-a^2)) : ... : ...).

        Best regards
        Sincerely
        Paul
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