SHARYGIN 2022 Problem 88

An isosceles trapezoid $ABCD$ ( $AB = CD$ ) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$ . Let $L$ be tha midpoint of $QD$ . Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$ .