Resumen de A Physical Proof of the Pythagorean Theorem
What proof of the Pythagorean theorem might appeal to a physics teacher? A proof that involved the notion of mass would surely be of interest. While various proofs of the Pythagorean theorem employ the circumcenter and incenter of a right-angled triangle,1 we are not aware of any proof that uses the triangle’s center of mass. This note details one such proof. Though far from the most elegant approach, we believe it to be novel.
Let us first summarize the required physics. Consider a region X ⊆ R2 of uniform density and total area A. If we decompose X into disjoint parts X1,…, Xn, then the center of mass can be found by first finding the center of mass (x⎯⎯x¯, y⎯⎯y¯) and area Aj of each part and then computing the weighted averages:
x⎯⎯=∑nj=1Ajx⎯⎯jAandy⎯⎯=∑nj=1Ajy⎯⎯jA.x¯=∑j=1nAjx¯jAandy¯=∑j=1nAjy¯jA.
We take for granted the well-known result that the center of mass of a triangle divides its medians in the ratio 2:1. We refer the reader to Ref. 2 for an entertaining and calculus-free proof of this fact. It follows that the centroid of a right-triangle with vertices (0,0),(a,0),(0,b) has coordinates (a/3,b/3). This is shown in Fig. 1.
