The Pythagorean Theorem - Educational Outreach

∂F/∂x = ∂F/∂y = 0x(C - x) 2 + y 2 = 0A 2 + B 2 = C 2F(x,y) = 0α + β = γFigure 2.38: Logically Equivalent Starting PointsStarting with any one of the five statements in Figure 2.38,any of the remaining four statements can be deduced viathe Cauliflower Proof by following a permissible path asindicated by the dashed lines. Each line is double-arrowedindicating total reversibility along that particular line.Once the Pythagorean Theorem is established, onecan show that the locus of pointsxcp[ C xcp] y2 cp0describes a circle centered at (C 2 ,0)with radiusC2 byrewriting the equation as2 2 2( xC2 ) [Ccp ycp 2] .This was not possible prior to the establishment of thePythagorean Theorem since the analytic equation for acircle is derived using the distance formula, a corollary ofthe Pythagorean Theorem. The dashed circle described by2 2 2( xC2 ) [Ccp ycp 2]in Figure 2.37 nicely reinforces the fact that LMN is aright triangle by the Inscribed Triangle Theorem of basicgeometry.79