Greatest Area of Quadrilateral with Sides in Arithmetic Sequence

From ProofWiki
Jump to navigation Jump to search


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 $d$.


$\AA = \sqrt {a b c d}$


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.


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