Butterflies in Hyperbola

Sidney H. Kung

We give an analytic proof of the Butterfly Theorem for hyperbolas.


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.


Through M, we introduce a new set of x-y-axes. Then the equation of the hyperbola is


(y + k)²



(1') - (y + k)² - b² = 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) (- m1²) - 2m1kx - (b² + k²) = 0.

The roots, x1 and x2, of equation (2) are the x-coordinates of C and D, where

x1 + x2=
- 2m1ka²

- m1²a²
- a²(b² + k²)

- m1²a²

Dividing these, we get


x1 + x2
- (b² + k²)


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


x3 + x4
- (b² + k²)


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


x1 + x2

x3 + x4

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

- x2x3

m1x2 - m2x3

m1x1 - m2x4

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


x1 - q
- y4

q - x4


q - x1

q - x4



q ≠ x1 and q ≠ x4.

Solving (4) for q, we have

(m1 - m2) x1x4

m1x1 - m2x4

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

(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.


  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. 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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