Bretschneider's Formula
Given a general quadrilateral with sides of lengths a, b, c, and d, the area is given by
(Coolidge 1939; Ivanov 1960; Beyer 1987, p. 123) where p and q are the diagonal lengths and s 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
![画像: K= sqrt((s-a)(s-b)(s-c)(s-d)-abcdcos^2[1/2(A+B)])](/index.cgi/speech/https://mathworld.wolfram.com/images/equations/BretschneidersFormula/NumberedEquation1.svg){kind=link}
(Bretschneider 1842; Strehlke 1842; Coolidge 1939; Beyer 1987, p. 123), where A and B are two opposite angles of the quadrilateral.
"Bretschneider's formula" can be derived by representing the sides of the quadrilateral by the vectors a, b, c, and d arranged such that a+b+c+d=0 and the diagonals by the vectors p and q arranged so that p=b+c and q=a+b. The area of a quadrilateral in terms of its diagonals is given by the two-dimensional cross product
| K=1/2|pxq|, |
(4)
|
which can be written
| K^2=1/4(pxq)·(pxq), |
(5)
|
where u·v 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, QuadrilateralExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Bretschneider's FormulaCite this as:
Weisstein, Eric W. "Bretschneider's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BretschneidersFormula.html