2021 AMC 12B Problem 11

Triangle ABC has AB=13,BC=14 and AC=15. Let P be the point on \overline{AC} such that PC=10. There are exactly two points D and E on line BP such that quadrilaterals ABCD and ABCE are trapezoids. What is the distance DE?

\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.