2NWing Butterfly Problem
This is an illustration of the 2NWing Butterfly problem that is discussed below.
What if applet does not run? 
2NWing Butterfly Theorem
Observe a 2Nwing butterfly cradle. Let M denote the nexus. Assume the spikes are crossed by the circles in points A_{N}, ..., A_{1}, (M), B_{1}, ..., B_{N} and C_{N}, ..., C_{1}, (M), D_{1}, ..., D_{N}, respectively. Let p be any permutation of the index set

Note that for N = 1 the theorem reduces to (A). The proof depends on two lemmas.
Lemma 1
Let in ΔRST, RU be a cevian through vertex R. Introduce angles a = ∠SRU and b = ∠URT. Then

The proof is a twoliner that follows from the identity
Area( ΔRST) = Area( ΔRSU) + Area( ΔRUT). 
Lemma 2 is a curious result in its own right.
Lemma 2
Assume three rays C, B, and Y emanate from point M with Y between C and B. Assume also that, for N > 1, points C_{1}, ..., C_{N} have been marked on C, while points B_{1}, ..., B_{N} have been marked on B. Let p be any permutation of indices 
The proof builds on Lemma 1 which is applied to each of the triangles MC_{i}B_{p(i)}:
(3)  sin(b + g)/MY_{i} = sin(g)/B_{p(i)} + sin(b)/MC_{i}, 
g and b are the angles CMY and YMB, respectively. Add up all the identities (3). After rearranging the terms, we'll get
(4)  sin(b + g)(1/MY_{1} + ... + 1/MY_{N}) = sin(g)(1/B_{1} + ... + 1/B_{N}) + sin(b)(1/MC_{1} + ... + 1/MC_{N}), 
in which the right hand side does not depend on the permutation p. The same therefore is true for the left hand side. 
Proof of Theorem
The proof easily follows from Lemma 2 by pidgin, or is it pigeon, induction (if a statement holds for
1/MS_{1} + ... + 1/MS_{N} = 1/MT_{1} + ... + 1/MT_{N}. 
But due to Lemma 2,
1/MS_{1} + ... + 1/MS_{N} = 1/MX_{1} + ... + 1/MX_{N}, while 1/MT_{1} + ... + 1/MT_{N} = 1/MY_{1} + ... + 1/MY_{N}, 
and the result follows. 
Remark
The theorem raises further questions. Each permutation can be represented as a product of irreducible cycles. It must be obvious that every such cycle is responsible for forming a unique butterfly. If the permutation is cyclic, the theorem yields a lone insect. However, in general, the N pairs of wings are split between a number of butterflies, one per an irreducible cycle that compose the permutation. This is an open question whether the butterflies are just stuck on top of each other and could be in principle separated, or whether their wings are so entangled that no separation is possible.
L. Bankoff was amazed with a fantastic variety of solutions to the Butterfly problem  "some by Desargues' theorem on involution, some by the use of cross ratios, others stemming from Menelaus, analytic geometry, trigonometry, advanced Euclidean geometry, and various other modalities ..." May the braids theory hold the answer to the problem of separation of the entangled butterflies.
Butterfly Theorem and Variants
 Butterfly theorem
 2NWing 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 Coaxial 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
Desargues' Theorem
 Desargues' Theorem
 2NWing Butterfly Problem
 Cevian Triangle
 Do You Speak Mathematics?
 Desargues in the Bride's Chair (with Pythagoras)
 Menelaus From Ceva
 Monge from Desargues
 Monge via Desargues
 Nobbs' Points, Gergonne Line
 Soddy Circles and David Eppstein's Centers
 Pascal Lines: Steiner and Kirkman Theorems II
 Pole and Polar with Respect to a Triangle
 Desargues' Hexagon
 The Lepidoptera of the Circles
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
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