6255Re: Some theorems on Miquel (not quadrilateral!) points
- Jan 3, 2003Dear Darij,
> >> > [DG] 2. Let Ma, Mb, Mc be the centers of the circles AB'C',collinear.
> >> > 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
> But they need not be! Therefore, the Baum theorem is a partialYes, you are perfectly right; I thought that you were talking about
> generalization of the Miquel circle in a quadrilateral, and not a
> Sorry for the possible-to-misunderstand use of the term "Miquel
the Miquel point of a complete quadrilateral.
With my apologizes for the misunderstanding. Jean-Pierre
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