ILL 1983 Problem 43

Given a square $ABCD$ , let $P, Q, R$ , and $S$ be four variable points on the sides $AB, BC, CD$ , and $DA$ , respectively. Determine the positions of the points $P, Q, R$ , and $S$ for which the quadrilateral $PQRS$ is a parallelogram, a rectangle, a square, or a trapezoid.