In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral.It works on both convex and concave quadrilaterals (but not crossed ones), whether it is cyclic or not.
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.
Bretschneider's formula is expressed as:
K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd ⋅ \cos2\left(
| \alpha+\gamma | |
| 2 |
\right)}
=\sqrt{(s-a)(s-b)(s-c)(s-d)-\tfrac{1}{2}abcd[1+\cos(\alpha+\gamma)]}.
\cos(\alpha+\gamma)=\cos(\beta+\delta)
\alpha+\beta+\gamma+\delta=360\circ.
Denote the area of the quadrilateral by . Then we have
\begin{align}K&=
| ad\sin\alpha | |
| 2 |
+
| bc\sin\gamma | |
| 2 |
.\end{align}
Therefore
2K=(ad)\sin\alpha+(bc)\sin\gamma.
4K2=(ad)2\sin2\alpha+(bc)2\sin2\gamma+2abcd\sin\alpha\sin\gamma.
The law of cosines implies that
a2+d2-2ad\cos\alpha=b2+c2-2bc\cos\gamma,
| (a2+d2-b2-c2)2 | |
| 4 |
=(ad)2\cos2\alpha+(bc)2\cos2\gamma-2abcd\cos\alpha\cos\gamma.
Adding this to the above formula for yields
\begin{align}4K2+
| (a2+d2-b2-c2)2 | |
| 4 |
&=(ad)2+(bc)2-2abcd\cos(\alpha+\gamma)\\ &=(ad+bc)2-2abcd-2abcd\cos(\alpha+\gamma)\\ &=(ad+bc)2-2abcd(\cos(\alpha+\gamma)+1)\\ &=(ad+bc)2-4abcd\left(
| \cos(\alpha+\gamma)+1 | |
| 2 |
\right)\\ &=(ad+bc)2-4abcd\cos2\left(
| \alpha+\gamma | |
| 2 |
\right).\end{align}
Note that:
| ||||
| \cos |
=
| 1+\cos(\alpha+\gamma) | |
| 2 |
| \alpha+\gamma | |
| 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)-16abcd\cos2\left(
| \alpha+\gamma | |
| 2 |
\right).
Introducing the semiperimeter
s=
| a+b+c+d | |
| 2 |
,
16K2=16(s-d)(s-c)(s-b)(s-a)-16abcd\cos2\left(
| \alpha+\gamma | |
| 2 |
\right)
K2=(s-a)(s-b)(s-c)(s-d)-abcd\cos2\left(
| \alpha+\gamma | |
| 2 |
\right)
and Bretschneider's formula follows after taking the square root of both sides:
K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd ⋅ \cos2\left(
| \alpha+\gamma | |
| 2 |
\right)}
The second form is given by using the cosine half-angle identity
\cos2\left(
| \alpha+\gamma | |
| 2 |
\right)=
| 1+\cos\left(\alpha+\gamma\right) | |
| 2 |
,
yielding
K=\sqrt{(s-a)(s-b)(s-c)(s-d)-\tfrac{1}{2}abcd[1+\cos(\alpha+\gamma)]}.
Emmanuel García has used the generalized half angle formulas to give an alternative proof. [1]
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 and to give[2] [3]
\begin{align} K&=\tfrac{1}{4}\sqrt{4e2f2-(b2+d2-a2-c2)2}\\ &=\sqrt{(s-a)(s-b)(s-c)(s-d)-\tfrac{1}{4}((ac+bd)2-e2f2)}\\ &=\sqrt{(s-a)(s-b)(s-c)(s-d)-\tfrac{1}{4}(ac+bd+ef)(ac+bd-ef)}\\ \end{align}