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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 from the right angle vertex to the hypotenuse. This subdivides the triangle c into two smaller triangles, a and b.

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The three triangles are similar. This means that the areas of the triangles $\Omega_a , \Omega_b , \Omega_c$ obey the relation

$\displaystyle \frac{a^2}{\Omega_a} = \frac{b^2}{\Omega_b} = \frac{c^2}{\Omega_c}$

Furthermore, the areas have the property

$\displaystyle \Omega_c = \Omega_a + \Omega_b$

Now from the first equation we can solve for $c^2$

$\displaystyle c^2 = \frac{a^2}{\Omega_a} \Omega_c$

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

$\displaystyle c^2 = \frac{a^2}{\Omega_a}\left(\Omega_a + \Omega_b\right)=a^2 +\frac{a^2}{\Omega_a}\Omega_b$

But we know from the first equation that

$\displaystyle \frac{a^2}{\Omega_a} = \frac{b^2}{\Omega_b}$

and thus

$\displaystyle c^2 = a^2 + b^2$

which is the famous result!