In the document Miquel2.html you can find a variant of the theorem, where one of the circles becomes a line. In the document Miquel_Point.html you can find another theorem on two intersecting lines.

The figure below shows a particular case, in which the centers of the four circles lie also on circle (c). Then the other intersection points build a rectangle. A clue fact is that with the angles indicated (z+(pi-y)/2)=(pi-x), which implies that {B,A*,O

Miquel_Point.html

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