Greatest Area of Quadrilateral with Sides in Arithmetic Sequence

Theorem
Let $Q$ be a quadrilateral whose sides $a$, $b$, $c$ and $d$ are in arithmetic sequence.

Let $\AA$ be the area of $Q$.

Let $Q$ be such that $\AA$ is the greatest area possible for one with sides $a$, $b$, $c$ and $d$.

Then:
 * $\AA = \sqrt {a b c d}$

Proof
We are given that $\AA$ is the greatest possible for a quadrilateral whose sides are $a$, $b$, $c$ and $d$.

From Area of Quadrilateral with Given Sides is Greatest when Quadrilateral is Cyclic, $Q$ is cyclic.

Hence $\AA$ can be found using Brahmagupta's Formula.

Let $s$ denote the semiperimeter of $Q$:


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

We are given that $a$, $b$, $c$ and $d$ are in arithmetic sequence.

, that means there exists $k$ such that:

where $k$ is the common difference.

Then:

and so: