Pythagorean Theorem (idea)

There are indeed many proofs of the Pythagorean Theorem. However, the following one is particularly charming. This proof was given by the 11 year old Albert Einstein.

Jacob Einstein taught his nephew Albert the fundamentals of Euclidean geometry. The 11 year old youngster felt that some of Euclid's proofs were unnecessarily complicated. For instance, proof of the Pythagorean theorem required many additional lines, angles and squares. The young Einstein came up with an elegant proof that only required one additional line, the altitude above the hypotenuse.

         *
        **   *
       * *       *      b
   a  *  *           *
     *   Ec              *
    *    *                   *
   *     *                       *
  *   Ea *       Eb                  *
 *       *                               *
* * * * * * * * * * * * * * * * * * * * * * *
               c

The height divides the large triangle into two smaller triangles that are similar to each other and similar to the large triangle. In Euclidean geometry, the area ratio of two similar closed figures is equal to the square of the ratio of corresponding linear dimensions. Therefore, the areas of the triangles E_a, E_b, and the larger E_c (E as in German Ebene) are:

E_a= ma²
E_b= mb²
E_c= mc²

(The resemblance with Einstein's famous relationship E=mc² is of course entirely coincidental).

The larger area E_c is the sum of the two smaller areas:

E_c=E_a+E_b

or:

mc² = ma²+mb²
c² = a²+ b²

Source: Fractals, Chaos, Power Laws, Minutes from an Infinite Paradise, Manfred Schroeder, New York, W.H. Freeman and Company, 1991