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453Re: What's notable about this point?

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  • F.Gremmen
    Mar 1, 2000
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      Clark Kimberling wrote on February 25, 2000
      >Would anyone care to find coordinates and nice properties of X (and other
      >incircle-inverses)?

      Dear Clark and all,

      Incircle-inverses are most of time complicated in barycentric coordinates
      as Paul Yiu already noted.
      Possibly the inverse of the Gergonne point or the Nagel point could
      work. The inverse of the Gergonne point has like X(187), inverse of
      the symmedian point in the circumcirle, reasonably simple barycentric
      coordinates w.r.t. ABC. They are

      (2*s*tan(A/2)^2 - a : 2*s*tan(B/2)^2 - b : 2*s*tan(C/2)^2 - c)
      (inverse of Gergonne point in the incircle)

      with a,b and c sidelengths and s = (a + b + c)/2.
      It doesn't occur in ETC.

      Barycentric coordinates for the inverse of the Nagel point in the incircle
      are (f(a,b,c) : f(b,c,a) : f(c,a,b))
      with f(a,b,c) equals the irreducible polynomial
      f(a,b,c) = -2*a^4 + (5*b + 5*c)*a^3 + (7*b^2 - 30*c*b + 7*c^2)*a^2
      + (-b^3 + 5*c*b^2 + 5*c^2*b - c^3)*a + (-b^4 + 2*c^2*b^2 - c^4)
      Not so easy to remember.

      Best Regards.

      Frans Gremmen, University of Nijmegen, The Netherlands.