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Midpoint of a segment with only circles

Source: flanders '05

September 28, 2005
ratiosymmetrygeometrycongruent triangles

Problem Statement

Starting with two points A and B, some circles and points are constructed as shown in the figure:[*]the circle with centre A through B [*]the circle with centre B through A [*]the circle with centre C through A [*]the circle with centre D through B [*]the circle with centre E through A [*]the circle with centre F through A [*]the circle with centre G through A (I think the wording is not very rigorous, you should assume intersections from the drawing) Show that MM is the midpoint of ABAB. http://www.mathlinks.ro/Forum/album_pic.php?pic_id=291