Let $ABCD$ be a rectangle in which $AB+BC+CD=20$ and $AE=9$, where $E$ is the midpoint of the side $BC$. Find the area of the rectangle.

Let the side lengths of the rectangle be $AB=x$ and $BC=y$.
Since $ABCD$ is a rectangle, $CD=AB=x$. The first condition gives:

Since $E$ is the midpoint of $BC$, we have $BE=y/2=10-x$.
In the right-angled triangle $ABE$, by the Pythagorean Theorem:

The area of the rectangle is $xy$:

From $2x^2-20x+19=0$, we have:

Thus, the area of the rectangle is exactly $19$.