6239Some theorems on Miquel points
- Jan 1, 2003Dear Floor van Lamoen, and Hyacinthos,
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
[Jacques Hadamard: Lecons de Geometrie Elementaire, exercise 344. I
don't know of a proof.]
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.]
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