# 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

- Butterfly theorem
- 2N-Wing Butterfly Theorem
- Better Butterfly Theorem
- Butterflies in Ellipse
- Butterflies in Hyperbola
- Butterflies in Quadrilaterals and Elsewhere
- Pinning Butterfly on Radical Axes
- Shearing Butterflies in Quadrilaterals
- The Plain Butterfly Theorem
- Two Butterflies Theorem
- Two Butterflies Theorem II
- Two Butterflies Theorem III
- Algebraic proof of the theorem of butterflies in quadrilaterals
- William Wallace's Proof of the Butterfly Theorem
- Butterfly theorem, a Projective Proof
- Areal Butterflies
- Butterflies in Similar Co-axial Conics
- Butterfly Trigonometry
- Butterfly in Kite
- Butterfly with Menelaus
- William Wallace's 1803 Statement of the Butterfly Theorem
- Butterfly in Inscriptible Quadrilateral
- Camouflaged Butterfly
- General Butterfly in Pictures
- Butterfly via Ceva
- Butterfly via the Scale Factor of the Wings
- Butterfly by Midline
- Stathis Koutras' Butterfly
- The Lepidoptera of the Circles
- The Lepidoptera of the Quadrilateral
- The Lepidoptera of the Quadrilateral II
- The Lepidoptera of the Triangle
- Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
- Two Butterfly Theorems by Sidney Kung
- 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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