Department of Mechanical Engineering, University of Applied Sciences, Dortmund, Germany.
Keywords: Brahmagupta's formula; vectorial proof; cyclic quadrilateral; geometry; maximum area;
In Euclidean geometry, Brahmagupta's formula calculates the aera A enclosed by a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The quadrilateral can be described by a loop closure of side vectors a, b, c, d running counter-clockwise (Fig.1) .
Brahmagupta's formula is used to determine the aera A of a cyclic quadrilateral given by its side lengths via 
with the semiperimeter
The area of a cyclic quadrilateral is the maximum possible for any quadrilateral with the given side lengths.
We start with the sum of the area of the two triangles in Fig.1 using half area product each .
Both angles in the triangles opposite to diagonal p sum up to α+γ=π and thus equals sinα=sinγ. So we can write in vectorial notation 
Substituting c~d in equation (3) using this relation and multiplying by 2 gives
Squaring both sides yields
and reusing Lagrange's Identity  (ab)2+(a~b)2=(ab)2 results in
Expressing the diagonal vector by its two triangle side vectors p=a+b=−(c+d) and squaring leads us to
The dot products ab and cd correspond to cosine's of their enclosed angles by cosα=−cosγ. In vectorial notation this now reads
Substituting cd in equation (5) resolves to
After squaring this expression and bringing it into this shape