Two Butterflies Theorem

Sidney H. Kung
June 2015

Theorem 1

Let $I\,$ be the point of intersection of two chords $AB\,$and $CD\,$ in a circle $(O).\,$ Construct a diameter $JK\,$ through $I\,$ that does not intersect either $AD\,$ or $BC\,$ so that points $J\,$ and $I\,$ lie on the opposite sides of chord $AC.\,$ Let $BA\cap CJ=M,\,$ $DC\cap AJ=N,\,$ $AC\cap JK=H.\,$

Two Butterfly Theorems by Sidney Kung

If the distances from points $A,\,$ $N\,$ $C,\,$ $M\,$ to the line $JK\,$ are $a,\,$ $n,\,$ $c,\,$ $m,\,$ respectively, then

$\displaystyle \frac{1}{a}+\frac{1}{n}=\frac{1}{c}+\frac{1}{m}.$

Proof of Theorem 1

Two Butterfly Theorems by Sidney Kung, Figure 1

With a reference to Figure 1, $\angle JHC=\angle KHA\,$ and $\angle JCH=\angle AKH.\,$ So, triangles $JCH\,$ and $AKH\,$ are similar. In particular,

(1)

$\displaystyle \frac{JC}{AK}=\frac{JH}{AH}.$

Triangles $JAH\,$ and $CKH\,$ are also similar. In particular,

(2)

$\displaystyle \frac{JA}{CK}=\frac{JH}{CH}.$

Dividing (1) by (2), we have

(3)

$\displaystyle \frac{JC}{AK}\cdot\frac{CK}{JA}=\frac{CH}{AH}.$

Observe that point i does not lie on any of the three side lines of $\Delta JCA;\,$ and, since $IA\cap CJ=M,\,$ $IC\cap AJ=N\,$ and $IJ\cap AC=H,\,$ then by Menelaus' theorem $(\Delta ACM,\,$ transversal $HJ;\,$ $\Delta AMJ,\,$ transversal $NC),$

(3')

$\displaystyle \frac{JM}{MC}\cdot\frac{CH}{HA}\cdot\frac{AN}{NJ}=1$

or,

(4)

$\displaystyle \frac{CM}{AN}\cdot\frac{JN}{JM}=\cdot\frac{CH}{AH}.$

Equating (3) and (4) gives

(5)

$\displaystyle \frac{JC}{AK}\cdot\frac{CK}{JA}=\frac{CM}{AN}\cdot\frac{JN}{JM}.$

In Figure 1, let $\angle AJK=\alpha,\,$ $\angle CJK=\beta.\,$ Then $CK=JK\cdot\sin\beta,\,$ $AK=JK\cdot\sin\alpha.\,$ Substituting into (5) and rearranging terms we get

$\displaystyle \frac{AN}{JA\cdot JN\cdot\sin\alpha}=\frac{CM}{JC\cdot JM\cdot\sin\beta}$

i.e., $\displaystyle \frac{JN+AJ}{JA\cdot JN\cdot\sin\alpha}=\frac{JM+JM}{JC\cdot JM\cdot\sin\beta},\,$ or

$\displaystyle \frac{1}{JA\cdot\sin\alpha}+\frac{1}{JN\cdot\sin\alpha}=\frac{1}{JC\cdot\sin\beta}+\frac{1}{JM\cdot\sin\beta}.$

Hence $\displaystyle \frac{1}{a}+\frac{1}{n}=\frac{1}{c}+\frac{1}{m}$ as required.

Theorem 2

Let $I\,$ be the point of intersection of two chords $AB\,$and $CD\,$ in a circle $(O).\,$ Construct a diameter $JK\,$ through $I\,$ that does not intersect either $AD\,$ or $BC\,$ so that points $J\,$ and $I\,$ lie on the same sides of chord $AC.\,$ Let $AI\cap JC=G,\,$ $CI\cap AJ=F,\,$ $JI\cap CA=H.\,$

Two Butterfly Theorems by Sidney Kung, theorem 2

If the distances from points $G,\,$ $C\,$ $F,\,$ $A\,$ to the line $JK\,$ are $g,\,$ $c,\,$ $f,\,$ $a,\,$ respectively, then

$\displaystyle \frac{1}{g}-\frac{1}{c}=\frac{1}{f}-\frac{1}{a}.$

Proof of Theorem 2

Two Butterfly Theorems by Sidney Kung, Figure 2

With a reference to Figure 2, $\Delta JAH\sim\Delta CKH,$ and $\Delta JCH\sim\Delta AKH.\,$ It follows that

(1)

$\displaystyle \frac{JA}{CK}=\frac{JH}{CH}.$


(2)

$\displaystyle \frac{JC}{AK}=\frac{JH}{AH}.$

Dividing (1) by (2), we have

(3)

$\displaystyle \frac{JC}{AK}\cdot\frac{CK}{JK}=\frac{CH}{AH}.$

In $\Delta JAC,\,$ the cevians $CF,\,$ $AG,\,$ and $JH\,$ are concurrent, thus, by Ceva's theorem,

(3')

$\displaystyle \frac{JF}{FA}\cdot\frac{AH}{HC}\cdot\frac{CG}{GJ}=1,$

or,

(4)

$\displaystyle \frac{CH}{AH}=\frac{JF}{FA}\cdot\frac{CG}{GJ}.$

Equating (3) and (4), we get

(5)

$\displaystyle \frac{JC}{AK}\cdot\frac{CK}{JA}=\frac{JF}{FA}\cdot\frac{CG}{GJ}.$

From Figure 2, $AK=JK\cdot\sin\alpha,\,$ $CK=JK\cdot\sin\beta.\,$ Substituting into (5) and rearranging terms, we have

$\displaystyle \frac{CG}{JC\cdot JG\cdot\sin\beta}=\frac{AF}{JA\cdot JF\cdot\sin\alpha},$

implying

$\displaystyle \frac{JC-JG}{JC\cdot JG\cdot\sin\beta}=\frac{JA-JF}{JA\cdot JF\cdot\sin\alpha},$

i.e.,

$\displaystyle \left(\frac{1}{JG}-\frac{1}{JC}\right)\cdot\frac{1}{\sin\beta}= \left(\frac{1}{JF}-\frac{1}{JA}\right)\cdot\frac{1}{\sin\alpha}.$

Hence, $\displaystyle \frac{1}{g}-\frac{1}{c}=\frac{1}{f}-\frac{1}{a},\,$ as required.

 

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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