Brahmagupta's Formula/Corollary

Corollary to Brahmagupta's Formula
The area of a cyclic quadrilateral with sides of lengths $a, b, c, d$ is:


 * $\dfrac{\sqrt{\left({a^2 + b^2 + c^2 + d^2}\right)^2 + 8 a b c d - 2 \left({a^4 + b^4 + c^4 + d^4}\right)}} 4$

Proof
Brahmagupta's Formula:


 * $\mathcal A = \sqrt{\left({s - a}\right) \left({s - b}\right) \left({s - c}\right) \left({s - d}\right)}$

where $s$ is the semiperimeter:


 * $s = \dfrac{a + b + c + d} 2$

Making the substitutions:

results in:
 * $\mathcal A = \dfrac{\sqrt{\left({a^2 + b^2 + c^2 + d^2}\right)^2 + 8 a b c d - 2 \left({a^4 + b^4 + c^4 + d^4}\right)}} 4$