Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC}$, with $BC<AD$ and $AB=CD$. There is a point $P$ in the plane such that $PA=2$, $PB=3$, and $PC=5$. If $\frac{BC}{AD}=\frac{1}{2}$, what is $PD$?