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

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• Jan 3, 2003

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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