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 A. 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 A = 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, this expression is equal to 16A = 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. 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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