The Pythagorean Theorem - Educational Outreach

2.7) Legendre Exploits Embedded SimilarityAdrian Marie Legendre was a well-known Frenchmathematician born at Toulouse in 1752. He died at Parisin 1833. Along with Lagrange and Laplace, Legendre can beconsidered on of the three fathers of modern analyticgeometry, a geometry that incorporates all the inherentpower of both algebra and calculus. With much of his life’swork devoted to the new analytic geometry, it should comeas no surprise that Legendre should be credited with apowerful, simple and thoroughly modern—for the time—new proof of the Pythagorean Proposition. Legendre’s proofstarts with the Windmill Light (Figure 2.11). Legendre thenpared it down to the diagram shown in Figure 2.25.axycbFigure 2.25: Legendre’s DiagramHe then demonstrated that the two right triangles formed bydropping a perpendicular from the vertex angle to thehypotenuse are both similar to the master triangle. Armedwith this knowledge, a little algebra finished the job.12x a a: x a c c22y b b: y b c c3:x y c ac22bc c a2 b2 cNotice that this is the first proof in our historical sequencelacking an obvious visual component.258