The Pythagorean Theorem - Educational Outreach

In addition, y serves as the altitude for the arbitrary triangle( , , ) defined by the construction shown in Figure2.33. The sum of the two square areas2A and2B inFigure 2.33 can be determined in terms of x and y asfollows:The terms2 222A B ( x y) ([ C x] y) 2Cy .2( x y) and([ C y2 x] ) are the areas of theleft and right outer enclosing squares, and the term2 Cy isthe combined area of the eight shaded triangles expressedas an equivalent rectangle. Define F ( x,y)as follows:F2 2 2( x,y) { A B C}2Then, substituting the expression for2A B2, we haveF(x,y) {( x y)F(x,y) {2xF(x,y) 4{ x[C x] y22 2y ([ C x] y)2 2Cx}2}222 2Cy C2}2We now restrict the function F to the compact, squaredomain D {( x,y)| 0 x C,0 y C}shown in Figure2.34 where the symbols BndD and IntD denote theboundary of D and interior of D respectively.(0, C) (C, C)BndDIntD(0, 0)(C, 0)Figure 2.34: Domain D of F72