Dear Darij

> 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.]

Using oriented angles of lines (modulo Pi)

<PXaX = <PXaA = <PC'A = <PC'B = <PXbB = <PXbX and Xa,Xb,P,X are

concyclic...

Happy New Year to all of you. Jean-Pierre