It often happens, in mathematics in particular, that in pursuit of one goal a person unexpectedly attains another. Serendipity is an occurrence of such a fortunate discovery, usually by accident. In the case at hand, I think that the chance played a minor role, if any at all. To discover a novel proof of the Pythagorean theorem, John Molokach needed and demonstrated uncommon perseverance and the power of observation.

*Theorem*

In a right triangle with legs a and b and hypotenuse c

*Proof*

John was solving the following problem:

There were two solution to the problem. One was based on the construction depicted below:

It is not hard to see (the main tools are the similarity of triangles and the formula for the sum of a geometric series), this construction leads to

The other - a more straightforward - solution was based on a simpler diagram:

and directly led to a pair of equations:

from which one obtains

Now, the "lucky part" was in John's insight to compare the expressions for m and n in the two solutions. Do that and observe that both comparisons lead to the Pythagorean formula.

At this point one may stop and congratulate John for a nice proof, but John himself did not stop to pat himself on a shoulder. He went on digging.

If \alpha is the angle opposite leg a, then from the first set of formulas,

By adding a second copy of the original triangle

and computing the area of the shaded portion in two ways one obtains

from which

The two expressions for \sin 2\alpha can, too, be equated, and - as before - the result is the Pythagorean identity.