Notes
three circles in a rectangle solution

Three Circles in a Rectangle

Solution by Similar Triangles

With the points labelled as above, triangles $O B A$ and $O C A$ are congruent and so $A C$ is the same length as $A B$, which is $5$. Triangles $O D C$ and $A D B$ are both right-angled and share the angle at $D$, so are similar.

Let $a$ be the length of $C D$ and $b$ of $O D$. The lengths of $O D C$ are, from shortest to longest, $1$, $a$, $b$ and of $A D B$ are $5$, $1 + b$, $5 + a$. So $1 + b = 5 a$ and $5 + a = 5 b$. Solving this gives $a = \frac{5}{12}$ and $b = \frac{13}{12}$. This means that $B D$ has length $\frac{25}{12}$.

Triangle $A F E$ is also similar to $A D B$. The length of $A E$ is $6$, so the length of $F E$ is $\frac{6}{5}$ths of the length of $D B$, so is $\frac{5}{2}$. The area of the rectangle is the $6 \times \frac{5}{2} = 15$.