Butterflies in Hyperbola
Sidney H. Kung
We give an analytic proof of the Butterfly Theorem for hyperbolas.
Theorem
Let M(0, k) be the midpoint of a chord AB parallel to the major axis of a hyperbola. Through M two other chords CD and EF are drawn. ED cuts AB at P and CF cuts AB at Q. Then M is also the midpoint of PQ.
Proof
Through M, we introduce a new set of xyaxes. Then the equation of the hyperbola is
(1) 

or,
(1')  b²x²  a²(y + k)²  a²b² = 0. 
Assume that the coordinates of the related points are as follows:
Substituting m_{1}x into (1') gives
(2)  (b²  a²m_{1}²) x²  2a²m_{1}kx  a² (b² + k²) = 0. 
The roots, x_{1} and x_{2}, of equation (2) are the xcoordinates of C and D, where
x_{1} + x_{2}  = 

x_{1} 

Dividing these, we get
 = 

Similarly, by substituting y = m_{2}x into (2) we will get
 = 

also (x_{3} and x_{4} are xcoordinates of E and F). Thus,
(*) 

which can further be rearranged (see note (I) below) to become
(3) 

Observe that C, Q, F are collinear. The slope of QC and that of FQ are equal. So,
 = 

Hence,
(4) 

q ≠ x_{1} and q ≠ x_{4}.
Solving (4) for q, we have
q  = 

In a similar manner, by equating the slopes of PE and DP we can get
p  = 

Now by comparing p and q, and taking into account (3), it is easy to see that
Note
Derivation of (3): crossmultiplying and rearranging,
m_{1}x_{1}x_{2}(x_{3} + x_{4}) = m_{2}x_{3}x_{4}(x_{1} + x_{2}) m_{1}x_{1}x_{2}x_{3}  m_{2}x_{3}x_{2}x_{4} = m_{2}x_{3}x_{4}x_{1}  m_{1}x_{1}x_{2}x_{4} x_{2}x_{3}(m_{1}x_{1}  m_{2}x_{4}) = x_{1}x_{4}(m_{2}x_{3}  m_{1}x_{2}), implying (3).
The above results are valid for parabolas. To verify, one may use the equation
x² = p(y + k), p, k > 0, to start.
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
 Butterfly in Complex Numbers
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Copyright © 19962018 Alexander Bogomolny
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