- Bretschneider's formula
In
geometry , Bretschneider's formula is the following expression for the area of aquadrilateral ,:Here, "p", "q", "r" and "s" are the sides of the quadrilateral, "T" is half the perimeter, and "A" and "C" are two opposite angles.Bretschneider's formula works on any quadrilateral regardless of whether it is cyclic or not.
Proof of Bretschneider's formula
Denote the area of the quadrilateral by "S". Then we have:
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.
Related formulas
Bretschneider's formula generalizes
Brahmagupta's formula for the area of acyclic quadrilateral , which in turn generalizesHeron's formula for the area of atriangle .External links
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