M ay 2 00 7 Pythagorean Triples and A New Pythagorean Theorem

Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational (integer) right triangles from a given one. 1. Harmonic and Symphonic Squares Consider an arbitrary triangle with altitude α corresponding to base β (see Figure 1a). Assuming that the base angles are acute, suppose that a square of side η is inscribed as shown in Figure 1b. Figure 1: Triangle and inscribed square Then α, β, η form a harmonic sum, i.e. satisfy (1). The equivalent formula η = αβ α+β is also convenient, and is found in some geometry books.