布雷特施奈德公式

其中， a,b,c,d 为四边形的边长， s 为半周长，即 s=(a+b+c+d)/2，而 α,γ 为其中二个对角。

此公式可用于任何四边形，不论是否为圆内接四边形，可视为婆罗摩笈多公式之推广。

布雷特施奈德公式的证明

设四边形的面积为 A： （公式略）

可推得： 4A²=(ad)²sin ²α +(bc)²sin ²γ +2abcdsin α sin γ

• 由余弦定理： a²+d²-2adcos α =b²+c²-2bccos γ
• 移项可得： （公式略）
• 加入 4A²： （公式略）

运用半角公式及因式分解可得： 16A²=(a+b+c-d)(a+b+d-c)(a+c+d-b)(b+c+d-a)-16abcd· cos ²((α +γ)/2)

代入半周长 s： 16A²=16(s-a)(s-b)(s-c)(s-d)-16abcd· cos ²((α +γ)/2)

∴ （公式略）

相关公式

只适用于圆内接四边形的婆罗摩笈多公式推广了适用于三角形面积的海伦公式，而布雷特施奈德公式推广了婆罗摩笈多公式。

布雷特施奈德公式中的三角函数修正项，可被重写成与四边形的二对角线长 p 及 q 有关的形式。 K=√((s-a)(s-b)(s-c)(s-d)-1/4(ac+bd+pq)(ac+bd-pq))

In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral.

It works on both convex and concave quadrilaterals, whether it is cyclic or not.

History

The German mathematician Carl Anton Bretschneider discovered the formula in 1842. The formula was also derived in the same year by the German mathematician Karl Georg Christian von Staudt.

Formulation

Bretschneider's formula is expressed as: K=√((s-a)(s-b)(s-c)(s-d)-abcd· cos ²((α +γ)/2))

= =√((s-a)(s-b)(s-c)(s-d)-1/2abcd[1+cos(α +γ)]).

Here, a, b, c, d are the sides of the quadrilateral, s is the semiperimeter, and α and γ are any two opposite angles. Bretschneider's formula works on any quadrilateral, whether it is cyclic or not.

Proof

Denote the area of the quadrilateral by K. Then we have （公式略）

Therefore 2K=(ad)sin α +(bc)sin γ.

4K²=(ad)²sin ²α +(bc)²sin ²γ +2abcdsin α sin γ.

The law of cosines implies that a²+d²-2adcos α =b²+c²-2bccos γ,

because both sides equal the square of the length of the diagonal BD. This can be rewritten as （公式略）

Adding this to the above formula for 4K2 yields （公式略）

Note that: cos ²(α +γ)/2=(1+cos(α +γ))/2 (a trigonometric identity true for all (α +γ)/2)

Following the same steps as in Brahmagupta's formula, this can be written as 16K²=(a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)-16abcdcos ²((α +γ)/2).

• Introducing the semiperimeter s=(a+b+c+d)/2,
• the above becomes 16K²=16(s-d)(s-c)(s-b)(s-a)-16abcdcos ²((α +γ)/2)
• K²=(s-a)(s-b)(s-c)(s-d)-abcdcos ²((α +γ)/2)

and Bretschneider's formula follows after taking the square root of both sides: K=√((s-a)(s-b)(s-c)(s-d)-abcd· cos ²((α +γ)/2))

The second form is given by using the cosine half-angle identity cos ²((α +γ)/2)=(1+cos (α +γ))/2,

yielding K=√((s-a)(s-b)(s-c)(s-d)-1/2abcd[1+cos(α +γ)]).

Emmanuel García has used the generalized half angle formulas to give an alternative proof.

Related formulae

Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.

The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals e and f to give （公式略）

References & further reading

Ayoub, Ayoub B. (2007). "Generalizations of Ptolemy and Brahmagupta Theorems". Mathematics and Computer Education. 41 (1). ISSN 0730-8639.

C. A. Bretschneider. Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes. Archiv der Mathematik und Physik, Band 2, 1842, S. 225-261 (online copy, German)

F. Strehlke: Zwei neue Sätze vom ebenen und sphärischen Viereck und Umkehrung des Ptolemaischen Lehrsatzes. Archiv der Mathematik und Physik, Band 2, 1842, S. 323-326 (online copy, German)