The Pythagorean Theorem - Educational Outreach

Note 1 : The reader is invited to compare the shearing proof of Pappusto the one presented in Section 2.11 and notice the sequencereversal.Note 2 : Pappus’ Theorem is restrictive in that one must construct thethird parallelogram according to the process laid out in the proof.Perhaps the theorem evolved after many trials with the squares ofPythagoras as Pappus tried to relate size of squares to the sides ofa general triangle. In doing so, squares became rectangles andparallelograms as the investigation broadened. I suspect, as isusually the case, perspiration and inspiration combined to producethe magnificent result above!Like Herron’s Theorem, both the PythagoreanTheorem and Pappus’ Theorem are independent and coequalresults. Figure 3.12 shows the methods associatedwith Pappus’ Theorem as they are used to prove thePythagorean Theorem, providing another example of aPythagorean shearing proof in addition to the one shown inSection 2.11. The second set of equally spaced sheer lineson the left side, are used to cut the smaller square into twopieces so that a fitting within the rails can quite literallyoccur.Double ShearAnd Fit sideA2A1A1A2A3Figure 3.12: Pappus Meets Pythagoras***110