The Pythagorean Theorem has several different proofs. Here is one interesting & an entirely different proof by the \(20^{th}\) President of the United States, **James Garfield**. He used three right triangles (ABE, AED, DEC) to make a trapezium.

Area of the trapezium \( [ABCD] \) | = \(\frac {1}{2}\) (sum of the lengths of parallel sides) \(\times\) height | \(= \frac {1}{2} (p+q) (p+q) = \frac {p^2+q^2+2pq}{2}\) |

Now, sum of the areas of the three triangles = \(\frac{pq}{2}+\frac{pq}{2}+\frac{r^2}{2}\)

Therefore, \(\frac{pq}{2}+\frac{pq}{2}+\frac{r^2}{2}\) \(= \frac {p^2+q^2+2pq}{2}\)

\(\Rightarrow 2pq+r^2 = p^2+q^2+2pq\)

\(\Rightarrow r^2 = p^2+q^2\)

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