Butterfly Trigonometry

Butterflies in quadrilaterals are not much different from butterflies living in circles. And the proof of their existence is readily reduced by affine transformation (shearing) to the case of orthodiagonal quadrilaterals. Two additional (trigonometric) proofs supplied by Sidney Kung (August 13, 2012) establish beyond any doubts the presence of butterflies in orthodiagonal (hence, in other, too) quadrilateral.

Theorem

Through the intersection \(O\) of the mutually perpendicular diagonals \(RS\), \(PQ\) of a convex quadrilateral \(RQSP\), draw two lines \(AB\) and \(CD\) that meet the sides of \(RQSP\) at \(A\), \(B\), \(C\), \(D\). If \(X = AD \cap PQ\), \(Y=CB\cap PQ\), and if \(OP = OQ\), then \(OX = OY\).

The first proof depends on a lemma proved elsewhere

Lemma 1

Let in \(\triangle RST\), \(RU\) be a cevian through vertex \(R\). Introduce angles \(\alpha = \angle SRU\) and \(\beta = \angle URT\). Then

\( \frac{\mbox{sin}(\alpha + \beta)}{RU} = \frac{\mbox{sin}(\alpha)}{RT} + \frac{\mbox{sin}(\beta)}{RS}. \)

Proof 1 of Theorem

Apply the lemma to triangles \(ROQ\), \(SOQ\), and \(BOC):

(1) \(\frac{1}{OC}=\frac{\mbox{sin}(\alpha)}{OR}+\frac{\mbox{cos}(\alpha)}{OQ},\)
(2) \(\frac{1}{OB}=\frac{\mbox{sin}(\beta)}{OS}+\frac{\mbox{cos}(\beta)}{OQ},\)
(3) \(\frac{\mbox{sin}(\alpha +\beta)}{OY}=\frac{\mbox{sin}(\beta)}{OC}+\frac{\mbox{sin}(\alpha)}{OB}.\)

Combining (1)-(3) we obtain

(4) \( \frac{\mbox{sin}(\alpha + \beta)}{OY}= \mbox{sin}(\beta)\left(\frac{OQ\cdot \mbox{sin}(\alpha)+OR\cdot \mbox{cos}(\alpha)}{OR\cdot OQ} \right)+ \mbox{sin}(\alpha)\left(\frac{OQ\cdot\mbox{sin}(\beta)+OS\cdot\mbox{cos}(\beta)}{OS\cdot OQ} \right). \)

Similarly, applying Lemma to triangles \(SOP\), \(ROP\), and \(AOD\), we have

(5) \(\frac{1}{OD}=\frac{\mbox{sin}(\alpha)}{OS}+\frac{\mbox{cos}(\alpha)}{OP},\)
(6) \(\frac{1}{OA}=\frac{\mbox{sin}(\beta)}{OR}+\frac{\mbox{cos}(\beta)}{OP},\)
(7) \(\frac{\mbox{sin}(\alpha +\beta)}{OX}=\frac{\mbox{sin}(\beta)}{OD}+\frac{\mbox{sin}(\alpha)}{OA}.\)

Combining (5)-(7) we obtain

(8) \( \frac{\mbox{sin}(\alpha + \beta)}{OX}= \mbox{sin}(\beta)\left(\frac{OP\cdot \mbox{sin}(\alpha)+OS\cdot \mbox{cos}(\alpha)}{OS\cdot OP} \right)+ \mbox{sin}(\alpha)\left(\frac{OP\cdot\mbox{sin}(\beta)+OR\cdot\mbox{cos}(\beta)}{OP\cdot OR} \right). \)

Remembering that \(OP=OQ\), a comparison of (4) and (8) shows that the right-hand sides are equal, and so are the left-hand sides, implying \(OX=OY\).

Proof 2

Let \(E\) and \(F\) be points symmetric to \(C\) and \(B\) with respect to the line \(RS\). Enlarging the left portion of the diagram

let \(AF\cap OP=Z\). \(F\) being a reflection of \(B\), line \(OP\) bisects \(\angle AOF\). Thus

(9) \( \frac{AF}{ZF}=\frac{AO}{OF}. \)

Note that triangles \(OFD\) and \(OPD\) are co-side (implying they have the same altitude from \(O\)); it follows that

(10) \( \frac{DF}{DP}=\frac{1/2\times OD\times OF\times\mbox{sin}(\theta)}{1/2\times OD\times OP\times\mbox{sin}(\alpha)}=\frac{OF\times \mbox{sin}(\theta)}{OP\times\mbox{sin}(\alpha)}. \)

Similarly, since triangles \(OEP\) and \(OEA\) share side \(OE\),

(11) \( \frac{PE}{EA}=\frac{OP\times \mbox{sin}(\alpha)}{OA\times\mbox{sin}(\theta)}. \)

Multiplying (9), (10), and (11) gives

\( \frac{AZ}{ZF}\frac{DF}{DP}\frac{PE}{EA}=\frac{AO}{OF}\frac{OF}{OP}\frac{OP}{OA}=1. \)

Thus, by the converse of Ceva's Theorem, \(AD\), \(OP\), and \(EF\) are concurrent. Since \(X\) and \(Y\) are on \(PQ\), and \(X\in EF\), \(Y\) must be the symmetric image of \(X\). So, \(OX=OY\).

References

  1. Sidney Kung, A butterfly theorem for quadrilaterals, Math. Mag. 78 (2005), 314

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

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