AIME 2005a Problem 15

Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$ Original wording of the problem: In $\triangle ABC$, $AB = 20$. The incircle of the triangle divides the median containing $C$ into three segments of equal length. Given that the area of $\triangle ABC$ is $m\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime, find $m + n$.