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A Proof of the Pythagorean Theorem From Heron's FormulaLet 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
RemarkFor 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. References
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