# Greatest Area of Quadrilateral with Sides in Arithmetic Sequence

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## 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 |

$\blacksquare$