Bretschneider's Formula (original) (raw)

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Given a general quadrilateral with sides of lengths ,,, and , the area is given by

(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)

(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) | | ------------------------------------------------------------------------------------------------------------ | --- |

which can be written

	(5)

where denotes a dot product. Making using of a vector quadruple product identity gives

But

Plugging this back in then gives the original formula (Ivanoff 1960).

See also

Brahmagupta's Formula, Heron's Formula, Quadrilateral

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.Bretschneider, C. A. "Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes." Archiv der Math. 2, 225-261, 1842.Coolidge, J. L. "A Historically Interesting Formula for the Area of a Quadrilateral." Amer. Math. Monthly 46, 345-347, 1939.Dostor, G. "Propriétés nouvelle du quadrilatère en général avec application aux quadrilatéres inscriptibles, circonscriptibles." Arch. Math. Phys. 48, 245-348, 1868.Hobson, E. W. A Treatise on Plane and Advanced Trigonometry. New York: Dover, pp. 204-205, 1957.Ivanoff, V. F. "Solution to Problem E1376: Bretschneider's Formula." Amer. Math. Monthly 67, 291-292, 1960.Strehlke, F. "Zwei neue Sätze vom ebenen und shparischen Viereck und Umkehrung des Ptolemaischen Lehrsatzes." Archiv der Math. 2, 33-326, 1842.

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Bretschneider's Formula

Cite this as:

Weisstein, Eric W. "Bretschneider's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BretschneidersFormula.html

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