Given a general quadrilateral with sides of lengths , , , and , the area is given by
(1)
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(2)
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(Coolidge 1939; Ivanov 1960; Beyer 1987, p. 123) where and are the diagonal lengths and is the semiperimeter. While this formula is termed Bretschneider's formula in Ivanoff (1960) and Beyer (1987, p. 123), this appears to be a misnomer. Coolidge (1939) gives the second form of this formula, stating "here is one [formula] which, so far as I can find out, is new," while at the same time crediting Bretschneider (1842) and Strehlke (1842) with "rather clumsy" proofs of the related formula
(3)
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(Bretschneider 1842; Strehlke 1842; Coolidge 1939; Beyer 1987, p. 123), where and are two opposite angles of the quadrilateral.
"Bretschneider's formula" can be derived by representing the sides of the quadrilateral by the vectors , , , and arranged such that and the diagonals by the vectors and arranged so that and . The area of a quadrilateral in terms of its diagonals is given by the two-dimensional cross product
(4)
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which can be written
(5)
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where denotes a dot product. Making using of a vector quadruple product identity gives
(6)
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(7)
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But
(8)
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(9)
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(10)
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(11)
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(12)
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Plugging this back in then gives the original formula (Ivanoff 1960).