## 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 