Difference between revisions of "Bretschneider's formula"
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<math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - ([(\vec{a} + \vec{c})\cdot(\vec{a} + \vec{c}) - \vec{a}\cdot\vec{a} - \vec{c}\cdot\vec{c}] - [(\vec{b} + \vec{d})\cdot(\vec{b} + \vec{d}) - \vec{b}\cdot\vec{b} - \vec{d}\cdot\vec{d}])^2} </math> | <math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - ([(\vec{a} + \vec{c})\cdot(\vec{a} + \vec{c}) - \vec{a}\cdot\vec{a} - \vec{c}\cdot\vec{c}] - [(\vec{b} + \vec{d})\cdot(\vec{b} + \vec{d}) - \vec{b}\cdot\vec{b} - \vec{d}\cdot\vec{d}])^2} </math> | ||
| − | <math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{ | + | <math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{c}|^2 + (\vec{a} + \vec{c})\cdot(\vec{a} + \vec{c}) - (\vec{b} + \vec{d})\cdot(\vec{b} + \vec{d})]^2} </math> |
| − | <math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{ | + | <math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{c}|^2 + |\vec{a} + \vec{c}|^2 - |\vec{b} + \vec{d}|^2]^2} </math> |
| − | <math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{ | + | <math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{c}|^2 + |\vec{a} + \vec{c}|^2 - |-(\vec{a} + \vec{c})|^2]^2} </math> |
| − | <math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{ | + | <math>= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{c}|^2]^2} </math> |
Then if <math>a, b, c, d</math> represent <math>|\vec{a}|, |\vec{b}|, |\vec{c}|, |\vec{d}|</math> (and are thus the side lengths) while <math>p, q</math> represent <math>|\vec{p}|, |\vec{q}|</math> (and are thus the diagonal lengths), the area of a quadrilateral is: | Then if <math>a, b, c, d</math> represent <math>|\vec{a}|, |\vec{b}|, |\vec{c}|, |\vec{d}|</math> (and are thus the side lengths) while <math>p, q</math> represent <math>|\vec{p}|, |\vec{q}|</math> (and are thus the diagonal lengths), the area of a quadrilateral is: | ||
Revision as of 14:58, 11 April 2014
Suppose we have a quadrilateral with edges of length 
(in that order) and diagonals of length 
. Bretschneider's formula states that the area
![$[ABCD]=\frac{1}{4}*\sqrt{4p^2q^2-(b^2+d^2-a^2-c^2)^2}$](http://latex.artofproblemsolving.com/7/5/9/7596249e635c730943d0f5493ab8e0d9f2399293.png)
.
It can be derived with vector geometry.
Proof
Suppose a quadrilateral has sides 
such that 
and that the diagonals of the quadrilateral are 
and 
. The area of any such quadrilateral is 
.
Lagrange's Identity states that 
. Therefore:
![$K^2 = \frac{1}{4} [ (\vec{p} \cdot \vec{p})(\vec{q} \cdot \vec{q}) - (\vec{p} \cdot \vec{q})(\vec{p} \cdot \vec{q}) ]$](http://latex.artofproblemsolving.com/b/6/d/b6d8cfebefae6108d1018fa1693d4b6c93510cfb.png)
![$= \frac{1}{4} [|\vec{p}|^2|\vec{q}|^2 - (\vec{p} \cdot \vec{q})^2]$](http://latex.artofproblemsolving.com/2/d/2/2d2c7fece433a2962e01e50598d03e30f64bce6b.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [2 (\vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b})]^2}$](http://latex.artofproblemsolving.com/9/c/c/9ccf2a4bc6104e21eeda53e7cbbb7b7d4fe7047b.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [2 \vec{b} \cdot (\vec{a} + \vec{b}) + 2 \vec{c} \cdot (\vec{a} + \vec{b})]^2}$](http://latex.artofproblemsolving.com/7/c/d/7cd10bed20a058d3d09fb72d3daedceaa3722307.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [-2 \vec{b} \cdot (\vec{c} + \vec{d}) + 2 \vec{c} \cdot (\vec{a} + \vec{b})]^2}$](http://latex.artofproblemsolving.com/7/e/b/7eb3346ccc404f33b51b30ea40611de912046b5b.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [-2 \vec{b} \cdot \vec{c} - 2 \vec{b} \cdot \vec{d} + 2 \vec{a} \cdot \vec{c} + 2 \vec{b} \cdot \vec{c}]^2}$](http://latex.artofproblemsolving.com/1/a/f/1af05bc30058c961a26aa5d285509ea384c1e01c.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [2 \vec{a} \cdot \vec{c} - 2 \vec{b} \cdot \vec{d}]^2}$](http://latex.artofproblemsolving.com/7/5/8/7588ef9fe40e58b9b4b50204f9f36cb37b06da53.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - ([(\vec{a} + \vec{c})\cdot(\vec{a} + \vec{c}) - \vec{a}\cdot\vec{a} - \vec{c}\cdot\vec{c}] - [(\vec{b} + \vec{d})\cdot(\vec{b} + \vec{d}) - \vec{b}\cdot\vec{b} - \vec{d}\cdot\vec{d}])^2}$](http://latex.artofproblemsolving.com/8/0/9/809d0ee15c6a549f26a9787ad3fa3108dd0fc1d3.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{c}|^2 + (\vec{a} + \vec{c})\cdot(\vec{a} + \vec{c}) - (\vec{b} + \vec{d})\cdot(\vec{b} + \vec{d})]^2}$](http://latex.artofproblemsolving.com/8/f/9/8f9a0e6892f0904d04edeccf6fc479f026971989.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{c}|^2 + |\vec{a} + \vec{c}|^2 - |\vec{b} + \vec{d}|^2]^2}$](http://latex.artofproblemsolving.com/e/f/8/ef812fd4a088fe06d41088b1b9352db61e9b9714.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{c}|^2 + |\vec{a} + \vec{c}|^2 - |-(\vec{a} + \vec{c})|^2]^2}$](http://latex.artofproblemsolving.com/d/9/3/d937b4aa44995ee24f87d0ef66ecdbc567bcfd16.png)
![$= \frac{1}{4} \sqrt{4 |\vec{p}|^2|\vec{q}|^2 - [|\vec{b}|^2 + |\vec{d}|^2 - |\vec{a}|^2 - |\vec{c}|^2]^2}$](http://latex.artofproblemsolving.com/a/3/a/a3a0661d14ddd63a4eff01539fbc07ac27044273.png)
Then if 
represent 
(and are thus the side lengths) while represent 
(and are thus the diagonal lengths), the area of a quadrilateral is:
![\[K = \frac{1}{4} \sqrt{4p^2q^2 - (b^2 + d^2 - a^2 - c^2)^2}\]](http://latex.artofproblemsolving.com/6/0/d/60dc4dbcb38afe751b3438020e7edec259fdf9a0.png)
See Also
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