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7409Thebault point

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  • jpehrmfr
    Aug 1, 2003
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      Dear Darij
      in Hyacinthos 7408, you wrote
      > Let ABC be a triangle and AA', BB', CC' its altitudes.
      > We have the following theorems:
      >
      > (1a) The Euler lines of triangles HB'C', HC'A', HA'B' concur
      > at one point, namely the Schiffler point of triangle
      > A'B'C'. It lies on the Euler line of triangle A'B'C'.
      >
      > (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.

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