Squares inside a triangle

Geometry Level 5

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 \(BC\)

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