Formula from Bretschneider

convex square

The formula of Bretschneider , named after Carl Anton Bretschneider , calculate the area of a rectangle based on its sides and diagonals. It is therefore a generalization of Brahmagupta's formula , which only applies to quadrilateral tendons and is itself a generalization of Heron's formula for the area of a triangle.

The area of a square ABCD with sides and diagonals is calculated as follows: ${\ displaystyle a, b, c, d}$ ${\ displaystyle e, f}$

{\ displaystyle {\ begin {aligned} F & = {\ tfrac {1} {4}} {\ sqrt {4e ^ {2} f ^ {2} - (b ^ {2} + d ^ {2} -a ^ {2} -c ^ {2}) ^ {2}}} \\ & = {\ sqrt {(sa) (sb) (sc) (sd) - {\ tfrac {1} {4}} (ac + bd + ef) (ac + bd-ef)}}. \ end {aligned}}}

Here half the circumference of the square is with and the correction term is 0, according to Ptolemy's theorem , if and only if it is a chordal square . ${\ displaystyle s}$ ${\ displaystyle s = {\ tfrac {1} {2}} (a + b + c + d)}$ ${\ displaystyle {\ tfrac {1} {4}} (ac + bd + ef) (ac + bd-ef)}$

The formula also has trigonometric variants in which two opposing interior angles of the square are used instead of the diagonals:

{\ displaystyle {\ begin {aligned} F & = {\ sqrt {(sa) (sb) (sc) (sd) -abcd \ cdot \ cos ^ {2} \ left ({\ frac {\ alpha + \ gamma} {2}} \ right)}} \\ & = {\ sqrt {(sa) (sb) (sc) (sd) - {\ tfrac {1} {2}} abcd [1+ \ cos (\ alpha + \ gamma)]}} \ end {aligned}}}

Here, too, the correction term is omitted in the special case of the chordal quadrilateral, since opposite angles complement each other in this and or apply. ${\ displaystyle \ pi}$ ${\ displaystyle \ cos ({\ tfrac {\ pi} {2}}) = 0}$ ${\ displaystyle \ cos (\ pi) = - 1}$

Both F. Strehlke and CA Bretschneider published trigonometric variants of the formula for the first time in 1842 in two separate articles, the first representation with the help of the diagonals appeared in a publication by G. Dostor (1868), while the second representation with the diagonals and the correction term in JL Coolidge (1939) goes back.

literature

Ayoub B. Ayoub: Generalizations of Ptolemy and Brahmagupta Theorems . In: Mathematics and Computer Education , Volume 41, Number 1, 2007, ISSN 0730-8639
VF Ivanoff, CF Pinzka, Joe Lipman: Solution to Problem E1376: Bretschneider's Formula . In: Amer. Math. Monthly , 67, 1960, pp. 291-292 ( JSTOR 2309706 )
JL Coolidge: A Historically Interesting Formula for the Area of a Quadrilateral . In: The American Mathematical Monthly , Vol. 46, No. 6 (June – July, 1939), pp. 345–347 ( JSTOR 2302891 )
Ernest William Hobson : A Treatise on Plane Trigonometry . Cambridge University Press, 1918, pp. 204–205 ( archive.org )
G. Dostor: Propriétés nouvelle du quadrilatère en général avec application aux quadrilatéres inscriptibles, circonscriptibles . In: Archive of Mathematics and Physics , Volume 48, 1868, pp. 245–348
CA Bretschneider. Investigation of the trigonometric relations of the rectilinear square . In: Archive of Mathematics and Physics , Volume 2, 1842, pp. 225–261 ( books.google.de )
F. Strehlke: Two new theorems of the plane and spherical square and the inversion of the Ptolemaic theorem . In: Archive of Mathematics and Physics , Volume 2, 1842, pp. 323–326 ( books.google.de )

Web links

Eric W. Weisstein : Bretschneider's Formula . In: MathWorld (English).
Bretschneider's formula on proofwiki.org