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:

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?

• Shearing Butterflies in Quadrilaterals
• Area of Parallelogram Formula by Shearing
• Parallelogram and Ellipses
• Proof 37 of the Pythagorean theorem - by David King
• Shearing a Polygon into a Triangle of Equal Area
• Pythagoras' Theorem By Sheer Shearing
• Shearing and Translation in Pythagorean Pants

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