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19636Re: A point related to a quadrilateral

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  • Chris Van Tienhoven
    Jan 1, 2011
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      Dear Jean-Pierre,

      > Consider a quadrilateral A_1,A_2,A_3,A_4
      > For k =1,2,3,4, T_k is the triangle with vertices the A_i except A_k; O_k is the circumcenter of T_k, (O_k) the circumcircle and B_k the isogonal conjugate of A_k wrt T_k
      > Then the inverse of B_k in (O_k) doesn't depend on k and this point M is the center of the homothecy mapping O_1,O_2,O_3,O_4 to B_1,B_2,B_3,B_4.
      > Is there a special name for this point M? Do you know some references?
      > Friendly. Jean-Pierre

      I do not know a special name for point M.
      However I noticed this. Maybe you know it already.
      Let T(A_1,A_2,A_3,A_4) = Transform A_1,A_2,A_3,A_4 --> O_1,O_2,O_3,O_4.
      Then T^2(A_1,A_2,A_3,A_4) produces a quadrilateral homethetic with A_1,A_2,A_3,A_4 only rotated 180 degrees. Again Center of Homothecy = M.
      T^4(A_1,A_2,A_3,A_4) produces a quadrilateral homothetic and with same orientation as A_1,A_2,A_3,A_4.

      Best regards and a creative year to all Hyacinthists!

      Chris van Tienhoven
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