The Pythagorean Theorem - Educational Outreach

Figure 2.32 is President Garfield’s Trapezoid diagram inupright position with its origin clearly linked to Figure 2.3.Recall that the area of a trapezoid, in particular the area ofthe trapezoid in Figure 2.32, is given by the formula:A Trap ( a b a b )12.Armed with this information, Garfield completes his proofwith a minimum of algebraic pen strokes as follows.121212121:A2:A3:aa1 ( a b) a b1 ( a b) a baa22222TrapTrap 2ab b 2ab b ab 12b2212ab 12b12222c ab 212c2set121212 aab ab ab 21212abc12122 bcc21222c2 c2121212ab ab ab Note: President Garfield actually published his proof in the 1876edition of the Journal of Education, Volume 3, Issue 161, where thetrapezoid is shown lying on its right side.It does not get any simpler than this! Garfield’s proof is amagnificent DRIII where all three fundamental quantities a,b, c are used in their natural and fundamental sense. Anextraordinary thing is that the proof was not discoveredsooner considering the ancient origins of Garfield’strapezoid. Isaac Newton, the co-inventor of calculus, oncesaid. “If I have seen further, it has been by standing on theshoulders of giants.” I am sure President Garfield, a giant inhis own right, would concur. Lastly, speaking of agreement,Garfield did have this to say about his extraordinary andsimple proof of the Pythagorean Proposition, “This is onething upon which Republicans and Democrats should bothagree.”.67