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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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