The Pythagorean Theorem - Educational Outreach

1 2y b b: y b c c2:A3:A4:Aunshadedsquareunshadedrectunshadedsquare b2bcA2c b2unshadedrectPutting the two pieces together (quite literally), we have:12 : A bigsquare c 2 2 2 2 2c a b a b cThe reader probably has discerned by now thatsimilarity arguments play a key role in many proofs of thePythagorean Theorem. This is indeed true. In fact, proof bysimilarity can be thought of a major subcategory just likeproof by dissection or proof by construction. Proof byvisualization is also a major subcategory requiring crystalclear,additive dissections in order to make the PythagoreanTheorem visually obvious without the help of analyticgeometry. Similarity proofs were first exploited in wholesalefashion by Legendre, a Frenchman that had the full powerof analytic geometry at his disposal. In Section 2.7, we willfurther reduce Euclid’s Windmill to its primal bare-bonesform via similarity as first exploited by Legendre.Note: More complicated proofs of the Pythagorean Theorem usuallyare a hybrid of several approaches. The proof just given can bethought of as a combination of construction, dissection, andsimilarity. Since similarity was the driving element in forming theargument, I would primarily characterize it as a similarity proof.Others may characteristic it as a construction proof since noargument is possible without the insertion of the perpendicularbisector. Nonetheless, creation of a perpendicular bisection creates adissection essential to the final addition of squares and rectangles!Bottom line: all things act together in concert.2.41