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
Ea, Eb, and the larger Ec (E as in
German Ebene) are:
Ea= ma2
Eb= mb2
Ec= mc2
(The resemblance with Einstein's famous relationship
E=mc2 is of course entirely coincidental).
The larger area Ec is the sum of the two smaller areas:
Ec=Ea+Eb
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