Pythagorean Theorem

0. Subdivide a right triangle into two smaller right triangles.

1. $$\alpha+\tau = 90 = \sigma+\tau$$ so $$\alpha = \sigma$$. Likewise $$\beta = \tau$$.
Hence the three triangles are similar.

2. By similarity,
$\frac{s}{a}=\frac{a}{c}, \quad \frac{t}{b} = \frac{b}{c}.$

3. By construction $$c = s + t$$.

4. Combining these, we have
$c = \frac{a^2}{c} + \frac{b^2}{c},$
therefore
$c^2 = a^2 + b^2.$

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