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Re: [EMHL] Circle of diametre [U;M] where U is the circumcenter

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  • Francois Rideau
    Dear Michel I suppose that coordinates or equations of all objects in triangle geometry were written as soon as barycentrics or trilinears were found, more
    Message 1 of 2 , Nov 29, 2008
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      Dear Michel

      I suppose that coordinates or equations of all objects in triangle geometry
      were written as soon as barycentrics or trilinears were found, more than
      150 years ago!
      As for your circles, it is easy to get equation of any circle <Gamma> in the
      triangle ABC plane.

      Name p(A), p(B), p(C) the respective powers of the vertices A, B, C wrt
      circle <Gamma>, then an equation of <Gamma> in barycentrics is:

      a²yz + b²zx + c²xy - (x + y + z)(p(A)x + p(B)y + p(C)z) = 0

      You get the coordinates of its center as the pole of the line at infinity.

      Notice that the line with equation:
      p(A)x + p(B)y + p(C)z = 0
      is the radical axis of <Gamma> and the ABC-circumcircle.

      Friendly
      Francois


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