Σάββατο, 3 Ιουλίου 2010

THREE CONCURRENT CIRCLES

Let 123 be a trianle, 4 a point, and 5,6,7 three points on the circles (423), (431),(412) resp. other than point 4.

Theorem:

The circles (167), (275) and (356) concur at a point (say) 8.


Special Cases:

1. Let the points 4,5,6,7 be concyclic (or collinear). See previous post.

2. Let the points 5,6,7 be the second intersections [= other than the point 4] of the lines 14, 24, 34 with the circles (423),(431),(412) resp.


3. Let the points 5,6,7 be the second intersections [= other than the point 4] of the circles (1,14),(2,24),(3,34) with the circles (423),(431), (412) resp.


4. Let 1',2',3' be the centers of the circles (423),(431),(412), resp. and the points 5,6,7, the second intersections [= other than the point 4] of the circles (42'3'),(43'1'),(41'2') with the circles (423),(431),(412) resp.


Continued 5

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