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.

Butterfly Theorem and Variants

1. Butterfly theorem
2. 2N-Wing Butterfly Theorem
3. Better Butterfly Theorem
4. The Lepidoptera of the Circles
5. The Lepidoptera of the Quadrilateral
6. The Lepidoptera of the Quadrilateral II
7. Butterflies in Ellipse
8. Butterflies in Hyperbola
9. Butterflies in Quadrilaterals and Elsewhere
10. Pinning Butterfly on Radical Axes
11. Shearing Butterflies in Quadrilaterals
12. The Plain Butterfly Theorem
13. Two Butterflies Theorem
14. Two Butterflies Theorem II
15. Two Butterflies Theorem III
16. Algebraic proof of the theorem of butterflies in quadrilaterals
17. William Wallace's Proof of the Butterfly Theorem
18. Butterfly theorem, a Projective Proof
19. Areal Butterflies
20. Butterflies in Similar Co-axial Conics
21. Butterfly Trigonometry
22. Butterfly in Kite
23. Butterfly with Menelaus
24. William Wallace's 1803 Statement of the Butterfly Theorem
25. Butterfly in Inscriptible Quadrilateral

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