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

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. The Lepidoptera of the Circles
5. The Lepidoptera of the Quadrilateral
6. The Lepidoptera of the Quadrilateral II
7. Butterflies in Ellipse
8. Butterflies in Hyperbola
9. Butterflies in Quadrilaterals and Elsewhere
10. Pinning Butterfly on Radical Axes
11. Shearing Butterflies in Quadrilaterals
12. The Plain Butterfly Theorem
13. Two Butterflies Theorem
14. Two Butterflies Theorem II
15. Two Butterflies Theorem III
16. Algebraic proof of the theorem of butterflies in quadrilaterals
17. William Wallace's Proof of the Butterfly Theorem
18. Butterfly theorem, a Projective Proof
19. Areal Butterflies
20. Butterflies in Similar Co-axial Conics
21. Butterfly Trigonometry
22. Butterfly in Kite
23. Butterfly with Menelaus
24. William Wallace's 1803 Statement of the Butterfly Theorem
25. Butterfly in Inscriptible Quadrilateral

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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