MathNet 2011 Question 57

Consider an isosceles trapezoid $ABCD$ with perpendicular diagonals. The parallel from the intersection point of the diagonals meets the non-parallel sides $[BC]$ and $[AD]$ at points $P$ and $R$ respectively. Point $Q$ is the mirror image of $P$ across the midpoint of $[BC]$. Show that

a) $QR = AD$;

b) $QR \perp AD$.