jmc

Triangle $ABC$ has side lengths $AB=12$, $BC=25$, and $CA=17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ=\omega$, the area of $PQRS$ can be expressed as the quadratic polynomial$$Area(PQRS)=\alpha \omega -\beta {\omega }^2.$$ Then the coefficient $\beta =\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.