The Pythagorean Theorem - Educational Outreach

3) Diamonds of the Same Mind“Philosophy is written in this grand book—I mean the universe—whichstands continually open to our gaze, but it cannot be understood unlessone first learns to comprehend the language and interpret the charactersin which it is written. It is written in the language of mathematics, andits characters are triangles, circles, and other geometric figures, withoutwhich it is humanly impossible to understand a single word of it;without these, one is wandering about in a dark labyrinth.” GalileoGalilei3.1) Extension to Similar AreasAll squares are geometrically similar. When we ‘fit’ length—the primary dimension associated with a square—to all2 2 2three sides of a right triangle, we have that a b c orA A by the Pythagorean Theorem. The equationA1 2A31A2 A3 not only applies to areas of squares fitted tothe sides of a right triangle, but also to any geometricallysimilar planar figure fitted to the sides of a right trianglesuch as the three similar crosses shown in Figure 3.1.A1acbA2A1acbA2A3A3Figure 3.1: Three Squares, Three Crosses88