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Proof of the Pythagorean Theorem

Theorem

For a right-angled triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)). This can be represented as: \[ c^2 = a^2 + b^2 \]

Proof

Consider two squares, each of side length \(a + b\). We can place four congruent right-angled triangles inside each square, with side lengths \(a\), \(b\), and \(c\).

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The Dowel Problem

Given that we have a circular dowel as such

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