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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

$\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

where

$s=\frac{a+b+c}{2}$

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,

$A = \sqrt{ \frac{1}{16} \begin{bmatrix}  0 & a^2 & b^2 & 1 \\  a^2 & 0 & c^2 & 1 \\ b^2 & c^2 & 0 & 1 \\  1 & 1 & 1 & 0  \end{bmatrix} }$, illustrates its similarity to Tartaglia's formula for the volume of a four-simplex.

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