A Proof of the Butterfly Theorem Using Ceva’s Theorem

Cesare Donolato
Forum Geometricorum, Volume 16 (2016) 185–186

Theorem (The Butterfly Theorem)

Through the midpoint $M\;$ of a chord $PQ\;$ of a circle, two other chords $AB\;$ and $CD\;$ are drawn. Chords $AD\;$ and $BC\;$ intersect $PQ\;$ at points $X\;$ and $Y,\;$ respectively.

Butterfly theorem

Then $M\;$ is also the midpoint of $XY.$

We introduce the points $A'\;$ and $D'\,$ that are the symmetric of $A\;$ and $D\;$ about $M,\;$ respectively. Hence, $MA' = MA\;$ and $MD' = MD.\;$ Next we connect the point $D'\;$ to $A'\;$ and $B,\;$ and call $E\;$ the intersection of $DB\;$ with the line through $P\;$ and $Q.\;$ Thus we have constructed triangle $MBD'\;$ with cevians $D'A',\;$ $ME,\;$ and $BC.\;$

Butterfly theorem, Donolato's lemma

We show that the segment $DA\;$ cuts the chord $PQ\;$ at the same point $Y\;$ as $BC,\;$ i.e., that the three cevians are concurrent at $Y.\;$ This property will be proved by applying Ceva’s theorem to triangle $MBD'.\;$

Lemma

In triangle $MBD,\;$ the cevians $DA,\;$ $ME,\;$ and $BC\;$ are concurrent at $Y.\;$

Proof of Lemma

We set $MA'=MA=\rho_1,\;$ $MB=r_1,\;$ $MC=r_2\;$ and $MD'=MD=\rho_2.\;$ We observe that $\displaystyle\frac{BE}{ED'}=\frac{r_1\sin\beta}{\rho_2\sin\alpha},\;$ the ratio of respective distances of $B\;$ and $D'\;$ from the line $PQ.\;$ Moreover, $A'B=r_1-\rho_1,\;$ and $D'C=\rho_2-r_2.\;$ Now,

(1)

$\displaystyle\begin{align} \frac{BE}{ED'}\cdot\frac{D'C}{CM}\cdot\frac{MA'}{A'B} &= \frac{r_1\sin\beta}{\rho_2\sin\alpha}\cdot\frac{\rho_2-r_2}{r_2}\cdot\frac{\rho_1}{r_1-\rho_1}\\ &=\frac{(\rho_2-r_2)\sin\beta}{(r_1-\rho_1)\sin\alpha}, \end{align}$

since $\rho_1r_1=\rho_2r_2\;$ by the Intersecting Chords Theorem.

The differences appearing in (1) can be written in terms of the distance $d = OM\;$ of the circle center $O\;$ to the chord $PQ,\;$ and the angles $\alpha\;$ and $\beta.\;$ The above figure shows that the projection of $OM\;$ onto the chord $PQ\;$ has length $d\sin\alpha,\;$ so that we get $\displaystyle\rho_2=\frac{1}{2}CD+d\sin\alpha,\;$ and $\displaystyle r_2=\frac{1}{2}C-d\sin\alpha.\;$ Hence, $\rho_2-r_2=2d\sin\alpha.\;$ Similarly, we find $r_1-\rho_1=2d\sin\beta.$ Substituting these expressions into (1) we obtain

$\displaystyle\frac{BE}{ED'}\cdot\frac{D'C}{CM}\cdot\frac{MA'}{A'B}=1.$

By Ceva's Theorem, the cevians $D'A',\;$ $ME,\;$ and $BC\;$ are concurrent. The common point is clearly $Y.$

Proof of Butterfly Theorem

We observe that triangle $MA'D'\;$ is congruent by construction to triangle $MAD,\;$ because two sides of the first $(MA',MD')\;$ are equal to two sides of the second $(MA,MD),\;$ and the included angles are equal. It follows that $\angle MD'Y = \angle MDX.\;$ Consequently, triangles $MD'Y\;$ and $MDX\;$ are also congruent, since they have equal two pairs of angles $(\angle MD'Y = \angle MDX,\;$ and $\angle YMD'= \angle XMD,\;$ vertical angles), as well as the included sides $(MD' = MD).\;$ This congruence implies that the corresponding sides $MY\;$ and $MX\;$ are equal. Therefore, $M\;$ is the midpoint of $XY\;$ and the butterfly theorem is proved.

Remark

It is rather interesting to compare this proof to William Wallace's 1803 Statement.

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
  35. Butterfly in Complex Numbers

|Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny

 63714315

Search by google: