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# Pythagorean Theorem Proof by an American President!

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, $$\frac{pq}{2}+\frac{pq}{2}+\frac{r^2}{2}$$ $$= \frac {p^2+q^2+2pq}{2}$$

$$\Rightarrow 2pq+r^2 = p^2+q^2+2pq$$

$$\Rightarrow r^2 = p^2+q^2$$

Note by Ameya Salankar
3 years, 1 month ago