ILL 1985 Problem 75

Let $ABCD$ be a rectangle, $AB = a, BC = b$ . Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being $d$ ) that are at an the angle $\phi, 0 \leq \phi \leq 90^{\circ},$ with respect to $AB$ . Let $L$ be the sum of the lengths of all the segments intersecting the rectangle. Find:

(a) how $L$ varies,

(b) a necessary and sufficient condition for $L$ to be a constant, and

(c) the value of this constant.