Two Butterflies Theorem II: What is it about?
A Mathematical Droodle
This generalization of the Two Butterflies theorem has been suggested by Nathan Bowler who observed that the two butterflies need not live on the same circle as long as the circles intersect. A further generalization appears to remove this restriction.
The two butterflies theorem may be seen as a statement on the properties of three elements: a circle, an inscribed quadrilateral, and a line with four collinear points. If there is one butterfly through the four points, there is infinitely many of them - a classical case of what is nowadays referred to as "porism." In this sense, the generalization to two circles raises a question: if a line with four collinear points leads to a porism in one circle, does it lead to a porism in any other circle? If the answer is positive, then the generalization is a direct consequence of the original statement.