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19705Re: [EMHL] A point related to a quadrilateral

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  • Barry Wolk
    Jan 9, 2011
      > Dear Jean-Pierre,
      > I am quite thrilled.
      > As always when I find something beautiful.
      > You described a point M depending on the 4 points A1, A2,
      > A3, A4 of a quadrilateral. Here is the structure behind it.
      > M = the inverse in the circumcircle of AiAjAk of the
      > isogonal conjugate of Al wrt AiAjAk,
      > where (i,j,k,l) is any permutation of (1,2,3,4).
      > Strangely enough this is true for each permutation.
      > There should be more properties/references relating to this point.
      > Best regards,
      > Chris van Tienhoven

      I did this using complex numbers as coordinates, where the formula for M should be symmetric in the 4 complex variables (A1,A2,A3,A4), and have some invariance properties. The answer can be expressed as follows:

      M is a quotient of two determinants. Each determinant is 4-by-4, and its rows correspond to the variables. In the numerator, the row for x is [1, x, x^2, x x#], where x# means the complex conjugate of x. In the denominator, the row for x is [1, x, x^2, x#].

      Can other ETC points be expressed in this way?
      Barry Wolk
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