Butterfly in Parabola
Experimental Mathematics

What Might This Be About?

10 July 2014, Created with GeoGebra

Theorem

Let $P(0,y_{0})$ be a point on the axis of a parabola $y=x^{2}.$ Through $P$ two lines $AB$ and $CD$ are drawn, with $A,B,C,D$ on parabola. $AC$ and $BD$ cut the line $y=y_{0}$ at $M$ and $N,$ respectively.

butterfly in parabola - problem

Then $PM=PN.$

The theorem is stated for the parabola $y=x^{2}$ but, since all parabolas are similar, it holds in the general case. The proof of the theorem depends on a lemma, see below.

Lemma

Given a parabola $y=x^{2}$ and three points $M(x_{0},y_{0}),$ $P(0,y_{0}),$ $N(-x_{0},y_{0}),$ $(y_{0}\gt 0, y_{0} \ne x_{0}^{2}).$ Line passing through $M$ intersects with the curve at $P_{1}(x_{1},y_{1})$ and $P_{2}(x_{2},y_{2});$ the extensions of $P_{1}P$ and $P_{2}P$ meet the curve at $P_{3}(x_{3}, y_{3})$ and $P_{4}(x_{4},y_{4}),$ respectively.

butterfly in parabola - lemma

Then $P_{3},$ $P_{4},$ and $N$ are collinear.

Proof of Lemma

Using the point-slope formula, we get the equation of $P_{1}P_{2}$:

(1)

$\displaystyle y=\frac{x_{2}^{2}-x_{1}^{2}}{x_{2}-x_{1}}(x-x_{1})+x_{1}^{2}=(x_{1}+x_{2})x-x_{1}x_{2}.$

Since $M$ is on $P_{1}P_{2}$ we have

(2)

$y_{0}=(x_{1}+x_{2})x_{0}-x_{1}x_{2}.$

The equation of $P_{1}P$ is, similarly, $\displaystyle y=\frac{x_{1}^{2}-y_{0}}{x_{1}}x+y_{0}.$ $P_{3}$ is the second intersection of $P_{1}P$ with the parabola, and can be found from $\displaystyle x^{2}=\frac{x_{1}^{2}-y_{0}}{x_{1}}x+y_{0},$ or

(3)

$x_{1}x^{2}-(x_{1}^{2}-y_{0})x-x_{1}y_{0}=0.$

The roots of (3) are the $x$-coordinates cf $P_{1}$ and $P_{3}$ so that $\displaystyle x_{1}+x_{3}= \frac{x_{1}^{2}-y_{0}}{x_{1}}$ and $x_{1}x_{3}=-y_{0}.$ So, $\displaystyle x_{3}=-\frac{y_{0}}{x_{1}}$ and $\displaystyle y_{3}=\big(\frac{y_{0}}{x_{1}}\big)^{2}.$ In a similar manner we get $\displaystyle x_{4}=-\frac{y_{0}}{x_{2}}$ and $\displaystyle y_{4}=\big(\frac{y_{0}}{x_{2}}\big)^{2}.$

The slope of $P_{3}P_{4}=m=\displaystyle\frac{y_{3}-y_{4}}{x_{3}-x_{4}}=-\frac{y_{0}(x_{1}+x_{2})}{x_{1}x_{2}}.$ Let $y=mx+b.$ Assuming it passes through $P_{4},$ yields $b=-\displaystyle\frac{y_{0}^{2}}{x_{1}x_{2}}.$ So, the equation of $P_{3}P_{4}$ is

(4)

$\displaystyle y=-\frac{y_{0}(x_{1}+x_{2})}{x_{1}x_{2}}x-\frac{y_{0}^{2}}{x_{1}x_{2}}.$

When $x=-x_{0}$ (4) yields $\displaystyle y=\frac{y_{0}}{x_{1}x_{2}}[x_{0}(x_{1}+x_{2})-y_{0}]=y_{0}$ (see (2)). Hence, $N$ lies on $P_{3}P_{4}$ and the three points $P_{3},$ $P_{4},$ $N$ are collinear.

Proof of Theorem

Let's denote $A=(x_{a},y_{a}),$ $B=(x_{b},y_{b}),$ $C=(x_{c},y_{c}),$ $D=(x_{d},y_{d}),$ $M=(p,y_{0}),$ $P=(0,y_{0}),$ $Q=(q,y_{0}).$

butterfly in parabola - problem

Let $y-y_{0}=k_{1}x$ and $y-y_{0}=k_{2}x$ be the equations of $AB$ and $CD,$ respectively. Since $A,$ $M,$ and $C$ are collinear,

$\displaystyle\frac{x_{a}-p}{x_{c}-p}=\frac{y_{a}-y_{0}}{y_{c}-y_{0}}=\frac{k_{1}x_{a}}{k_{2}x_{c}}.$

Solve this for $p$ to get

(5)

$\displaystyle p=\frac{(k_{1}-k_{2})x_{a}x_{c}}{k_{1}x_{a}-k_{2}x_{c}}.$

Similarly, since $B,$ $N,$ and $D$ are collinear,

(6)

$\displaystyle q=\frac{(k_{1}-k_{2})x_{b}x_{d}}{k_{1}x_{b}-k_{2}x_{d}}.$

Substituting the equation $y-y_{0}=k_{1}x$ of $AB$ into $y=x^{2}$ we have $x^{2}-k_{1}x-y_{0}=0,$ , whose roots are the $x$-coordinates of $A$ and $B,$ and $-y_{0}=x_{a}x_{b}$ and $k_{1}=x_{a}+x_{b}.$ So, $\displaystyle\frac{k_{1}x_{a}x_{b}}{x_{a}+x_{b}}=-y_{0}.$ Similarly, $\displaystyle\frac{k_{2}x_{c}x_{d}}{x_{c}+x_{d}}=-y_{0}.$ Equating the two gives

$k_{1}(x_{c}+x_{d})x_{a}x_{b}=k_{2}(x_{a}+x_{b})x_{c}x_{d}.$

This is equivalent to

$\displaystyle\frac{x_{a}x_{c}}{k_{1}x_{a}-k_{2}x_{c}}=-\frac{x_{b}x_{d}}{k_{1}x_{b}-k_{2}x_{d}}.$

By combining this result with (5) and (6) we see that $p=-q.$ Therefore, $PM=PN.$

Acknowledgment

The theorem and its proof are due to Sidney H. Kung.

Conic Sections > Parabola

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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