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6255Re: Some theorems on Miquel (not quadrilateral!) points

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  • jpehrmfr <jean-pierre.ehrmann@wanadoo.fr>
    Jan 3, 2003
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      Dear Darij,

      > >> > [DG] 2. Let Ma, Mb, Mc be the centers of the circles AB'C',
      > >> > BC'A', CA'B' [it is well-known that triangles MaMbMc and
      > >> > ABC are similar], let M' be the circumcenter of triangle
      > >> > MaMbMc, and M the circumcenter of triangle ABC. Then
      > >> > M'M = M'P.
      > >> >
      > >> > [This is a theorem by Peter Baum; proposed in the little
      > >> > German periodical "Die Wurzel" (see the website
      > >> > wurzel.org), in the 12/98 issue. Solved by Sefket
      > >> > Arslanagic in the 5/99 issue; the solution was quite
      > >> > involved.]
      > >>
      > >> [JPE] It is well known (Steiner 1827) that Ma,Mb,Mc,M,P are
      > >> concyclic (Miquel cirle of the quadrilateral). Hence, this
      > >> theorem is not a recent one.
      >
      > If I understand correctly, you assume that A', B', C' are
      collinear.
      > But they need not be! Therefore, the Baum theorem is a partial
      > generalization of the Miquel circle in a quadrilateral, and not a
      > corollary.
      >
      > Sorry for the possible-to-misunderstand use of the term "Miquel
      > point".

      Yes, you are perfectly right; I thought that you were talking about
      the Miquel point of a complete quadrilateral.
      With my apologizes for the misunderstanding. Jean-Pierre
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