Let $D$ be the centre of circle which passes through the mid-points of the three edges of $\triangle ABC$, $E$ and $F$ be points on $AB$ and $AC$ respectively such that $DE\perp AB$ and $DF\perp AC$. It is known that $AB = 15$, $AC = 20$, $4DE = 3DF$. $BC^2 = q+r\sqrt{s}$, where $q$ and $r$ are integers and $s$ is a square free integer. Enter $q + r + s$.