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.

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