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.

Without loss of generality, that means there exists $k$ such that:

\(\ds b\)

\(=\)

\(\ds a + k\)

\(\ds c\)

\(=\)

\(\ds a + 2 k\)

\(\ds d\)

\(=\)

\(\ds a + 3 k\)

where $k$ is the common difference.

Then:

\(\ds s\)

\(=\)

\(\ds \dfrac {a + b + c + d} 2\)

Definition of Semiperimeter

\(\ds \)

\(=\)

\(\ds \dfrac {a + \paren {a + k} + \paren {a + 2 k} + \paren {a + 3 k} } 2\)

\(\ds \)

\(=\)

\(\ds \dfrac {4 a + 6 k} 2\)

\(\text {(1)}: \quad\)

\(\ds \)

\(=\)

\(\ds 2 a + 3 k\)

and so:

\(\ds \AA\)

\(=\)

\(\ds \sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d} }\)

Brahmagupta's Formula

\(\ds \)

\(=\)

\(\ds \sqrt {\paren {a + 3 k} \times \paren {a + 2 k} \times \paren {a + k} \times a}\)

substituting $s = 2 a + 3 k$ from $(1)$ and simplifying

\(\ds \)

\(=\)

\(\ds \sqrt {a b c d}\)

from above

$\blacksquare$