The Pythagorean Theorem - Educational Outreach

2.11) Shear, Shape, and AreaOur last major category of proof for the PythagoreanProposition is that of a shearing proof. Shearing proofs havebeen around for at least one thousand years, but they haveincreased in popularity with the advent of the moderncomputer and associated computer graphics.The heart of a shearing proof is a rectangle thatchanges to a parallelogram preserving area as shown inFigure 2.40. Area is preserved as long as the length andaltitude remain the same. In a sense, one could say that ashearing force F is needed to alter the shape of therectangle into the associated parallelogram, hence the nameshearing proof. Shearing proofs distinguish themselves fromtransformer proofs in that playing pieces will undergo bothshape changes and position changes. In transformer proofs,the playing pieces only undergo position changes.FbA1A2hFA3Identical b and hthroughout impliesA1=A2=A3Figure 2.40: Shearing a RectangleShearing proofs are primarily visual in nature, making themfantastic to watch when animated on a computer. As such,they are of special interest to mathematical hobbyists, whocollectively maintain their delightful pursuit of new ones.83