The Pythagorean Theorem is one of the most important discoveries in mathematics. Here is a simple derivation (or "proof") of the theorem. We start with a large right triangle, labeled c. Draw a perpendicular line from the right angle vertex to the hypotenuse. This subdivides the triangle c into two smaller triangles, a and b. The three triangles are similar. This means that the areas of the triangles obey the relation
Furthermore, the areas have the property
From the first equation we can solve for
Using the second equation, the can be replaced
We know from the first equation that
which is the famous result!