Stathis Koutras' Butterfly

What Is This About?

Problem

Stathis Koutras' Butterfly, problem

Solution

Let $EM,FN\,$ be the orthogonal projections of $SO\,$ on $AD,BC,\,$ respectively. And let $T\,$ be the intersection of $AD\,$ and $BC.\,$

Stathis Koutras' Butterfly, proof

Since $OS\perp PQ,\,$ we may apply Stathis Koutras' theorem to obtain

(1)

$\displaystyle\frac{EM}{FN}=\frac{TQ}{TP}.$

But $EM,FN\,$ are the corresponding segments of obviously similar triangles $SAD\,$ and $SBC\,$ ($E,F\,$ are the feet of the altitudes whereas $M,N\,$ the midpoints of the corresponding sides) and, therefore, their ratio shall be equal to the ratio of the corresponding altitudes, i.e., $\displaystyle\frac{EM}{FN}=\frac{SE}{SF}.\,$ Combining this with (1),

$\displaystyle\frac{TQ}{TP}=\frac{SE}{SF}\,\Rightarrow\,TQ\cdot SF=TP\cdot SE\,\Rightarrow\,[\Delta TSQ]=[\Delta TSP],$

where $[X]\,$ denotes the area of shape $X.\,$ The two triangles share the altitude from $T,\,$ implying that $S\,$ is indeed the midpoint of $PQ,\,$ thus proving the Butterfly Theorem.

Acknowledgment

Stathis Koutras has kindly posted at the CutTheKnotMath facebook page his novel proof of the Butterfly Theorem.

 

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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Copyright © 1996-2017 Alexander Bogomolny

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