An important theorem in plane geometry, also known as Hero's formula. Given the lengths of the sides 
,
, and 
and the semiperimeter
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(1)
|
of a triangle, Heron's formula gives the area 
of the triangle
as
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(2)
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Heron's formula may be stated beautifully using a Cayley-Menger determinant as
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(3)
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Another highly symmetrical form is given by
![(4Delta)^2=[a^2 b^2 c^2][-1 1 1; 1 -1 1; 1 1 -1][a^2; b^2; c^2]](/images/equations/HeronsFormula/NumberedEquation4.svg) |
(4)
|
(Buchholz 1992).
Expressing the side lengths 
, 
,
and 
in terms of the radii 
,
, and 
' of the mutually tangent circles centered on the triangle
vertices (which define the Soddy circles),
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(5)
|
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(6)
|
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(7)
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gives the particularly pretty form
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(8)
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Heron's proof (Dunham 1990) is ingenious but extremely convoluted, bringing together a sequence of apparently unrelated geometric identities and relying on the properties of cyclic quadrilaterals and right triangles. Heron's proof can be found in Proposition 1.8 of his work Metrica (ca. 100 BC-100 AD). This manuscript had been lost for centuries until a fragment was discovered in 1894 and a complete copy in 1896 (Dunham 1990, p. 118). More recently, writings of the Arab scholar Abu'l Raihan Muhammed al-Biruni have credited the formula to Heron's predecessor Archimedes prior to 212 BC (van der Waerden 1961, pp. 228 and 277; Coxeter and Greitzer 1967, p. 59; Kline 1990; Bell 1986, p. 58; Dunham 1990, p. 127).
A much more accessible algebraic proof proceeds from the law of cosines,
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(9)
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Then
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(10)
|
giving
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(11)
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(12)
|
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(13)
|
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(14)
|
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(15)
|
(Coxeter 1969). Heron's formula contains the Pythagorean theorem as a degenerate case.
