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6242Re: Some theorems on Miquel points

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  • jpehrmfr <jean-pierre.ehrmann@wanadoo.fr>
    Jan 1, 2003
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      Dear Darij

      > Since we have started to investigate Miquel points, here are
      > two "forgotten" theorems on them.
      >
      > The Miquel point configuration is defined as follows:
      >
      > On the sidelines BC, CA, AB of a triangle ABC, points A', B', C'
      are
      > chosen and the circles AB'C', BC'A', CA'B' are drawn. Then, as we
      > know, these circles have a common point P (the Miquel point).
      >
      > Now the two theorems:
      >
      > 1. If X is an arbitrary point in the plane, then the second points
      of
      > intersection of the lines XA, XB, XC with the circles AB'C',
      BC'A',
      > CA'B' (the first intersections being A, B, C) are concyclic with P
      > and X.
      >
      > [Jacques Hadamard: Lecons de Geometrie Elementaire, exercise 344.
      I
      > don't know of a proof.]

      Using oriented angles of lines (modulo Pi)
      <PXaX = <PXaA = <PC'A = <PC'B = <PXbB = <PXbX and Xa,Xb,P,X are
      concyclic...
      Happy New Year to all of you. Jean-Pierre
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