Cyclic quadrilaterals & Brahmagupta’s formula

I suppose every reader of this ‘ere blog will know Heron’s formula for the area $latex K$ of a triangle with sides $latex a,b,c$:

$latex K = \sqrt{s(s-a)(s-b)(s-c)}$

where $latex s$ is the “semi-perimeter”:

$latex \displaystyle{s=\frac{a+b+c}{2}.}$

The formula is not at all hard to prove: see the Wikipedia page for two elementary proofs.

However, I have only recently become aware of Brahmagupta’s formula for the area of a cyclic quadrilateral. A cyclic quadrilateral, if you didn’t know, is a (convex) quadrilateral all of whose points lie on a circle:

And if the edges have lengths $latex a,b,c,d$ as shown, then the formula states that the area is given by

$latex K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$

where as above $latex s$ is the semi-perimeter:

$latex \displaystyle{s=\frac{a+b+c+d}{2}.}$

This can be seen to be a generalization of Heron’s formula. Although the formula is named for Brahmagupta (598 – 670), who does indeed seem to…