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10977RE: [EMHL] circumcevian reflections

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  • Floor en Lyanne van Lamoen
    Jan 6, 2005
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      Dear Bernard,

      > > [FvL]To start the new year let me offer a little theorem:
      > >
      > > If A'B'C' is a circumcevian triangle [of point P], and A"B"C" are the
      > > reflections of
      > > A'B'C' through the sides of ABC, then the circles (ABC), (A"B"C),
      > > (A"BC")
      > > and (AB"C") are concurrent in one point.

      [BG]
      > your point is the isogonal conjugate of the infinite point of the
      > direction which is perpendicular to HP.
      >
      > notice that ABC and A"B"C" are perspective iff P lies on the Jerabek
      > hyperbola or at infinity.

      Thanks, Bernard!

      The locus of the perspector for P on the Jerabek hyperbola is a quartic, the
      isogonal conjugate of the conic

      SUM b^4c^4SA(b^2-c^2)^4x^2 + a^6b^2c^2(c^2-a^2)^2(a^2-b^2)^2yz = 0
      cyclic

      Kind regards,
      Floor.
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