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Heron's formula
In geometry, Heron's formula states that the area of a triangle whose sides have lengths a, b, c is
- <math>\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}</math>
where
- <math>s=\frac{a+b+c}{2}</math>
This formula is credited to Heron of Alexandria, although it is possible that it may have been known long before Heron's time.
The formula is in fact a special case of Brahmagupta's formula for the area of a cyclic quadrilateral; both of which are special cases of Bretschneider's formula[?] for the area of a quadrilateral.
Expressing Heron's formula with a determinant in terms of the squares of the distances between the three given vertices,
- <math> A = \sqrt{ \frac{1}{16} \begin{bmatrix}
0 & a^2 & b^2 & 1 \\
a^2 & 0 & c^2 & 1 \\
b^2 & c^2 & 0 & 1 \\
1 & 1 & 1 & 0
\end{bmatrix} } </math>,
illustrates its similarity to Tartaglia's formula for the volume of a four-simplex.
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