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.