Dear 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

and X.

[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.]

Darij Grinberg