Shearing Butterflies in Quadrilaterals

Shearing is an affine transformation f: P → f(P) such that, for any point P, the line through P and f(P) is parallel to a fixed line, say λ, while the distance from P to f(P) is proportional to the distance from P to λ. All points on λ itself remain fixed: f(P) = P. Shearing transform relates to Euclid propositions I.35 - I.38 that assert preservation of areas of parallelograms and triangles with fixed base and other vertices moved parallel to it. Shearing is the main tool in several proofs of the Pythagorean theorem.

In a Cartesian system of coordinates where λ serves as the x-axis, the shearing transform has the matrix form

so that v = y and u = x + ky.

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 1/IM - 1/IA = 1/IN - 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.

Copyright © 1996-2009 Alexander Bogomolny