# 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

- Butterfly theorem
- 2N-Wing Butterfly Theorem
- Better Butterfly Theorem
- Butterflies in Ellipse
- Butterflies in Hyperbola
- Butterflies in Quadrilaterals and Elsewhere
- Pinning Butterfly on Radical Axes
- Shearing Butterflies in Quadrilaterals
- The Plain Butterfly Theorem
- Two Butterflies Theorem
- Two Butterflies Theorem II
- Two Butterflies Theorem III
- Algebraic proof of the theorem of butterflies in quadrilaterals
- William Wallace's Proof of the Butterfly Theorem
- Butterfly theorem, a Projective Proof
- Areal Butterflies
- Butterflies in Similar Co-axial Conics
- Butterfly Trigonometry
- Butterfly in Kite
- Butterfly with Menelaus
- William Wallace's 1803 Statement of the Butterfly Theorem
- Butterfly in Inscriptible Quadrilateral
- Camouflaged Butterfly
- General Butterfly in Pictures
- Butterfly via Ceva
- Butterfly via the Scale Factor of the Wings
- Butterfly by Midline
- Stathis Koutras' Butterfly
- The Lepidoptera of the Circles
- The Lepidoptera of the Quadrilateral
- The Lepidoptera of the Quadrilateral II
- The Lepidoptera of the Triangle
- Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
- Two Butterfly Theorems by Sidney Kung

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