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 x-y-axes. 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 m1x into (1') gives
(2) | (b² - a²m1²) x² - 2a²m1kx - a² (b² + k²) = 0. |
The roots, x1 and x2, of equation (2) are the x-coordinates of C and D, where
x1 + x2 | = |
|
x1 |
|
Dividing these, we get
| = |
|
Similarly, by substituting y = m2x into (2) we will get
| = |
|
also (x3 and x4 are x-coordinates 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) |
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q ≠ x1 and q ≠ x4.
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): cross-multiplying and rearranging,
m1x1x2(x3 + x4) = m2x3x4(x1 + x2) m1x1x2x3 - m2x3x2x4 = m2x3x4x1 - m1x1x2x4 x2x3(m1x1 - m2x4) = x1x4(m2x3 - m1x2), 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
- 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
- Butterfly in Complex Numbers
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Copyright © 1996-2018 Alexander Bogomolny
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