6242Re: Some theorems on Miquel points
- Jan 1, 2003Dear Darij
> Since we have started to investigate Miquel points, here areare
> 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'
> chosen and the circles AB'C', BC'A', CA'B' are drawn. Then, as weof
> 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
> 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 PI
> and X.
> [Jacques Hadamard: Lecons de Geometrie Elementaire, exercise 344.
> 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
Happy New Year to all of you. Jean-Pierre
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