Difference between revisions of "Bretschneider's formula"
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* [[Geometry]] | * [[Geometry]] | ||
− | {{ | + | ==The Proof== |
+ | Denote the area of the quadrilateral by ''S''. Then we have | ||
+ | :<math> \begin{align} S &= \text{area of } \triangle ADB + \text{area of } \triangle BDC \ | ||
+ | &= \tfrac{1}{2}pq\sin A + \tfrac{1}{2}rs\sin C | ||
+ | \end{align} </math> | ||
+ | |||
+ | Therefore | ||
+ | :<math> 4S^2 = (pq)^2\sin^2 A + (rs)^2\sin^2 C + 2pqrs\sin A\sin C. \, </math> | ||
+ | |||
+ | The [[cosine law]] implies that | ||
+ | :<math> p^2 + q^2 -2pq\cos A = r^2 + s^2 -2rs\cos C, \, </math> | ||
+ | because both sides equal the square of the length of the diagonal ''BD''. This can be rewritten as | ||
+ | :<math>\tfrac14 (r^2 + s^2 - p^2 - q^2)^2 = (pq)^2\cos^2 A +(rs)^2\cos^2 C -2 pqrs\cos A\cos C. \,</math> | ||
+ | |||
+ | Substituting this in the above formula for <math>4S^2</math> yields | ||
+ | :<math>4S^2 + \tfrac14 (r^2 + s^2 - p^2 - q^2)^2 = (pq)^2 + (rs)^2 - 2pqrs\cos (A+C). \, </math> | ||
+ | |||
+ | This can be written as | ||
+ | :<math>16S^2 = (r+s+p-q)(r+s+q-p)(r+p+q-s)(s+p+q-r) - 16pqrs \cos^2 \frac{A+C}2. </math> | ||
+ | |||
+ | Introducing the semiperimeter | ||
+ | :<math>T = \frac{p+q+r+s}{2},</math> | ||
+ | the above becomes | ||
+ | :<math>16S^2 = 16(T-p)(T-q)(T-r)(T-s) - 16pqrs \cos^2 \frac{A+C}2</math> | ||
+ | and Bretschneider's formula follows. | ||
+ | NOTE TO ALL: this proof was taken from Wikipedia on December the 1st, 2006. |
Revision as of 22:56, 1 December 2006
Suppose we have a quadrilateral with edges of length (in that order) and diagonals of length . Bretschneider's formula states that the area .
It can be derived with vector geometry.
See Also
The Proof
Denote the area of the quadrilateral by S. Then we have
- $\begin{align} S &= \text{area of } \triangle ADB + \text{area of } \triangle BDC \
&= \tfrac{1}{2}pq\sin A + \tfrac{1}{2}rs\sin C
\end{align}$ (Error compiling LaTeX. Unknown error_msg)
Therefore
The cosine law implies that
because both sides equal the square of the length of the diagonal BD. This can be rewritten as
Substituting this in the above formula for yields
This can be written as
Introducing the semiperimeter
the above becomes
and Bretschneider's formula follows. NOTE TO ALL: this proof was taken from Wikipedia on December the 1st, 2006.