Squares inside a triangle

Given triangle $ABC$ with $BC=115, \sin A=\frac {5}{7}, \frac {AB}{AC}=\frac {3}{5}$ . There is a set of points $P$ on segment $BC$ satisfying the following:

For each $P$ there exists a square $PQRS$ such that $Q,S$ lie on $\overline {AC},\overline {AB}$ respectively, and $R$ lies inside triangle $ABC$ .

Find the length of the segment(s) that these points $P$ make up on $BC$