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7410Re: Thebault point

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  • Darij Grinberg
    Aug 1, 2003
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      Dear Jean-Pierre Ehrmann,

      Thanks for the mail. You wrote:

      >> > (1b) The Euler lines of triangles AB'C', BC'A', CA'B' concur
      >> > at one point on the nine-point circle of triangle ABC.
      >> >
      >> > Note that (1a) follows from the definition of the Schiffler
      >> > point (H is the incenter of triangle A'B'C')
      >>
      >> Note that if an angle of ABC is obtuse, the incenter of A'B'C' is
      >> not H, but the corresponding vertex of ABC.

      Yes, of course I know this, the excenters of a triangle
      also lie on the Neuberg cubic.

      >> > and (1b) is a result of Victor Thebault.
      >> Notethat the common point is the center of Jerabek hyperbola.

      I have met this before. The point M where the Euler lines
      of triangles AB'C', BC'A', CA'B' meet has the property
      that one of the equations MA' = MB' + MC' or cyclically
      holds. This was stated by Thebault. Can anybody find a
      SYNTHETIC proof?

      Thanks,
      Sincerely,
      Darij Grinberg
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