AMC 12 2022a Problem 20

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 $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$

$\textbf{(A) }\frac{1}{4} \textbf{(B) }\frac{1}{3} \textbf{(C) }\frac{1}{2} \textbf{(D) }\frac{2}{3} \textbf{(E) }\frac{3}{4}$