A Proof of the Pythagorean Theorem From Heron's Formula

Let the sides of a triangle have lengths a,b and c. Introduce the semiperimeter p = (a + b + c)/2 and the area S. Then Heron's formula asserts that

W. Dunham analyzes the original Heron's proof in his Journey through Genius.

For the right triangle with hypotenuse c, we have S = ab/2. We'll modify the right hand side of the formula by noting that

It takes a little algebra to show that

For the right triangle, 16S² = 4a²b². So we have

Taking all terms to the left side and grouping them yields

With a little more effort

And finally

Remark

For a quadrilateral with sides a, b, c and d inscribed in a circle there exists a generalization of Heron's formula discovered by Brahmagupta. In this case, the semiperimeter is defined as p = (a + b + c + d)/2. Then the following formula holds

Since any triangle is inscribable in a circle, we may let one side, say d, shrink to 0. This leads to Heron's formula.