Pythagorean Theorem

Pythagorean Theorem is surely one of the most popular theorems in mathematics, which says that in a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

Pythagorean Theorem: $c^2 = a^2 + b^2$.

This theorem has so many proofs. One of the proofs is rather interesting because it was due to James Abram Garfield - the 20th President of the United States.

President Garfield's proof is very simple. It relies on finding the area of the following trapezoid in two different ways.

This trapezoid has two bases of length $a$ and $b$. The height of the trapezoid is $a + b$. So the area of the trapezoid is $$\frac{1}{2}(a+b)(a+b) = \frac{1}{2}(a^2 + b^2 + 2ab).$$

The other way to calculate the area of the trapezoid is to sum up the three areas of the triangles, which is $$\frac{1}{2}ab + \frac{1}{2} c^2 + \frac{1}{2}ab = \frac{1}{2}(c^2 + 2ab).$$

Comparing the two results we obtain the Pythagorean Theorem! $$c^2 = a^2 + b^2.$$

Integer numbers $a$, $b$, $c$ that satisfy the equation $$c^2 = a^2 + b^2$$ are called the Pythagorean triples. You can read more about Pythagorean triples in this post.

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