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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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