Shearing Butterflies in Quadrilaterals

Shearing - one of affine transformations - belongs to every geometry problem solver tool chest. Jacopo (Jack) D'Aurizio has observed that shearing can be used to prove the Butterfly theorem in a quadrilateral:

Through the intersection \(I\) of the diagonals \(AC\), \(BD\) of a convex quadrilateral \(ABCD\), draw two lines \(EF\) and \(HG\) that meet the sides of \(ABCD\) in \(E\), \(F\), \(G\), \(H\). Let \(M\) and \(N\) be the intersections of \(EG\) and \(FH\) with \(AC\). Then

\(\frac{1}{IM} - \frac{1}{IA} = \frac{1}{IN} - \frac{1}{IC}\).

Any shearing parallel to the diagonal \(AC\) leaves points on \(AC\) fixed and, therefore, does not affect the required equality \(\frac{1}{IM} - \frac{1}{IA} = \frac{1}{IN} - \frac{1}{IC}\).

What if applet does not run?

Applying a shearing transformation, Jack first shows that the Butterfly theorem holds in any quadrilateral provided it holds in the orthodiagonal ones. For the latter, he applies an inversion with center \(I\). Any such inversion fixes the lines through \(I\), although not pointwise. In particular, the diagonals \(AC\) and \(BD\) remain fixed and the same holds for the lines \(EF\) and \(GH\). The inversion transforms the sides of the quadrilateral into circles through \(I\) giving a set of four circles with perpendicular radical axes. In this configuration, Jack employs some trigonometry. For the details, check the pdf file.

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

What Is Shear Transform?

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