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. 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
  26. Camouflaged Butterfly
  27. Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
  28. Butterfly via Ceva
  29. Butterfly via the Scale Factor of the Wings
  30. Butterfly by Midline
  31. Stathis Koutras' Butterfly
  32. General Butterfly in Pictures
  33. Two Butterfly Theorems by Sidney Kung

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