## 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 $\Omega_a , \Omega_b , \Omega_c$ obey the relation

Furthermore, the areas have the property

From the first equation we can solve for $c^2:$

Using the second equation, the $\Omega_c$ can be replaced

We know from the first equation that

Thus,

which is the famous result!