The Pythagorean Theorem - Educational Outreach

But, this is precisely the nature of algebra and analyticgeometry where abstract ideas are more precisely (andabstractly) conveyed than by descriptive (visual) geometryalone. The downside is that visual intuition plays a minimalrole as similarity arguments produce the result via a fewalgebraic pen strokes. Thus, this is not a suitable beginner’sproof.Similarity proofs have been presented throughoutthis chapter, but Legendre’s is historically the absoluteminimum in terms of both geometric augmentation (thedrawing of additional construction lines, etc.) and algebraicterseness. Thus, it is included as a major milestone in oursurvey of Pythagorean proofs. To summarize, Legendre’sproof can be characterized as an embedded similarity proofwhere two smaller triangles are created by the dropping ofjust one perpendicular from the vertex of the mastertriangle. All three triangles—master and the two created—are mutually similar. Algebra and similarity principlescomplete the argument in a masterful and modern way.Note: as a dissection proof, Legendre’s proof could be characterizedas a DRII, but the visual dissection is useless without the powerfulhelp of algebra, essential to the completion of the argument.Not all similarity proofs rely on complicated ratiossuch as2x a / c to evaluate constructed lineardimensions in terms of the three primary quantities a, b, &c. Figure 2.26 is the diagram recently used (2002) by J.Barry Sutton to prove the Pythagorean Proposition usingsimilarity principles with minimally altered primaryquantities.EAac-bBcbbCbDCircle of radius bFigure 2.26 Barry Sutton’s Diagram59