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)	x² a² - (y + k)² b² = 1

or,

(1')	b²x² - a²(y + k)² - a²b² = 0.

Assume that the coordinates of the related points are as follows: M(0, 0), C(x₁, y₁), D(x₂, y₂), E(x₃, y₃), F(x₄, y₄), P(p, 0), Q(q, 0); and that the equations of lines DC and EF are y = m₁x and y = m₂x, respectively.

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

=

- - 2m1ka² b² - m1²a²

    and    

x1	x2 = - a²(b² + k²) b² - m1²a²

Dividing these, we get

m1x1x2 x1 + x2

=

- (b² + k²) 2k

Similarly, by substituting y = m₂x into (2) we will get

m2x3x4 x3 + x4

=

- (b² + k²) 2k

also (x₃ and x₄ are x-coordinates of E and F). Thus,

(*)	m1x1x2 x1 + x2 = m2x3x4 x3 + x4

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

(3)	- x2x3 m1x2 - m2x3 = x1x4 m1x1 - m2x4

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

y1 x1 - q

=

- y4 q - x4

Hence,

(4)	q - x1 q - x4 = y1 y4 = m1x1 m2x4

q ≠ x₁ and q ≠ x₄.

Solving (4) for q, we have

q

=

(m1 - m2) x1x4 m1x1 - m2x4

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

p

=

(m1 - m2) x2x3 m1x2 - m2x3

Now by comparing p and q, and taking into account (3), it is easy to see that |p| = |q|. Therefore, MP = MQ.

Note

1. 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).

2. 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

1. Butterfly theorem
2. 2N-Wing Butterfly Theorem
3. Better Butterfly Theorem
4. The Lepidoptera of the Circles
5. The Lepidoptera of the Quadrilateral
6. The Lepidoptera of the Quadrilateral II
7. Butterflies in Ellipse
8. Butterflies in Hyperbola
9. Butterflies in Quadrilaterals and Elsewhere
10. Pinning Butterfly on Radical Axes
11. Shearing Butterflies in Quadrilaterals
12. The Plain Butterfly Theorem
13. Two Butterflies Theorem
14. Two Butterflies Theorem II
15. Two Butterflies Theorem III
16. Algebraic proof of the theorem of butterflies in quadrilaterals
17. William Wallace's Proof of the Butterfly Theorem
18. Butterfly theorem, a Projective Proof
19. Areal Butterflies
20. Butterflies in Similar Co-axial Conics
21. Butterfly Trigonometry
22. Butterfly in Kite
23. Butterfly with Menelaus
24. William Wallace's 1803 Statement of the Butterfly Theorem
25. Butterfly in Inscriptible Quadrilateral

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