Two Butterflies Theorem II: What is it about?
A Mathematical Droodle

p>21 July 2015, Created with GeoGebra

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. Butterflies in Ellipse
  5. Butterflies in Hyperbola
  6. Butterflies in Quadrilaterals and Elsewhere
  7. Pinning Butterfly on Radical Axes
  8. Shearing Butterflies in Quadrilaterals
  9. The Plain Butterfly Theorem
  10. Two Butterflies Theorem
  11. Two Butterflies Theorem II
  12. Two Butterflies Theorem III
  13. Algebraic proof of the theorem of butterflies in quadrilaterals
  14. William Wallace's Proof of the Butterfly Theorem
  15. Butterfly theorem, a Projective Proof
  16. Areal Butterflies
  17. Butterflies in Similar Co-axial Conics
  18. Butterfly Trigonometry
  19. Butterfly in Kite
  20. Butterfly with Menelaus
  21. William Wallace's 1803 Statement of the Butterfly Theorem
  22. Butterfly in Inscriptible Quadrilateral
  23. Camouflaged Butterfly
  24. General Butterfly in Pictures
  25. Butterfly via Ceva
  26. Butterfly via the Scale Factor of the Wings
  27. Butterfly by Midline
  28. Stathis Koutras' Butterfly
  29. The Lepidoptera of the Circles
  30. The Lepidoptera of the Quadrilateral
  31. The Lepidoptera of the Quadrilateral II
  32. The Lepidoptera of the Triangle
  33. Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
  34. Two Butterfly Theorems by Sidney Kung

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