Heron's formula

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Heron's formula gives the area of a triangle when the lengths of all three sides are known. Given the lengths of the three sides of a triangle, a, b, c, the area of the triangle can be determined as

A = \sqrt{s(s-a)(s-b)(s-c)}

where s = \dfrac{a+b+c}{2} , semi-perimeter of the triangle.

The formula can be transformed to the following interesting forms.


A = \sqrt{s(s-a)(s-b)(s-c)}

\quad= \dfrac{1}{4}\sqrt{(a+b+c)(a+b-c)(a+c-b)(b+c-a)}

\quad= \dfrac{1}{4}\sqrt{[(a+b)+c)][(a+b)-c][c+(a-b)][(c-(a-b)]}

\quad=\dfrac{1}{4}\sqrt{[(a+b)^2-c^2][c^2-(a-b)^2]}

\quad=\dfrac{1}{4}\sqrt{(2ab+a^2+b^2-c^2)(2ab-a^2-b^2+c^2)}

\quad=\dfrac{1}{4}\sqrt{4a^2b^2-(a^2+b^2-c^2)}

\quad=\dfrac{1}{4}\sqrt{2(a^2b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}

\quad=\dfrac{1}{4}\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}


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