Difference between revisions of "Bretschneider's formula"
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Suppose we have a [[quadrilateral]] with [[edge]]s of length <math>a,b,c,d</math> (in that order) and [[diagonal]]s of length <math>p, q</math>. '''Bretschneider's formula''' states that the [[area]] | Suppose we have a [[quadrilateral]] with [[edge]]s of length <math>a,b,c,d</math> (in that order) and [[diagonal]]s of length <math>p, q</math>. '''Bretschneider's formula''' states that the [[area]] | ||
− | <math>[ABCD]=\frac{1}{4} | + | <math>[ABCD]=\frac{1}{4} \cdot \sqrt{4p^2q^2-(b^2+d^2-a^2-c^2)^2}</math>. |
It can be derived with [[vector]] [[geometry]]. | It can be derived with [[vector]] [[geometry]]. | ||
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<math>K = \frac{1}{2} |\vec{p} \times \vec{q}| </math> | <math>K = \frac{1}{2} |\vec{p} \times \vec{q}| </math> | ||
− | + | [[Lagrange's Identity]] states that <math>|\vec{a}|^2|\vec{b}|^2-(\vec{a}\cdot\vec{b})^2=|\vec{a}\times\vec{b}|^2 \implies \sqrt{|\vec{a}|^2|\vec{b}|^2-(\vec{a}\cdot\vec{b})^2}=|\vec{a}\times\vec{b}|</math>. Therefore: | |
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− | [[Lagrange's Identity]] states that <math> | ||
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<math>K = \frac{1}{2} \sqrt{|\vec{p}|^2|\vec{q}|^2 - (\vec{p} \cdot \vec{q})^2} </math> | <math>K = \frac{1}{2} \sqrt{|\vec{p}|^2|\vec{q}|^2 - (\vec{p} \cdot \vec{q})^2} </math> | ||
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* [[Geometry]] | * [[Geometry]] | ||
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[[Category:Geometry]] | [[Category:Geometry]] | ||
[[Category:Theorems]] | [[Category:Theorems]] |
Latest revision as of 03:51, 12 February 2021
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.
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:
Then if represent
(and are thus the side lengths) while
represent
(and are thus the diagonal lengths), the area of a quadrilateral is: