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6254Re: Some theorems on Miquel (not quadrilateral!) points

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  • Darij Grinberg <darij_grinberg@web.de>
    Jan 3, 2003
      (Answer to Hyacinthos message #6240)

      In message #6240, Jean-Pierre Ehrmann wrote:

      >> > [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".

      Sincerely,
      Darij Grinberg
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