## Pythagorean Theorem

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

Thus,

which is the famous result!