Question 3
In the plane, let $ABCD$ be a square of side $2$, with $A,B,C,D$ in cyclic order. The segment $AB$ is moved continuously until it coincides with $CD$, with $A$ ending at $C$ and $B$ at $D$. Let $S$ be the area swept out by $AB$ during the motion. Prove that a motion can be found for which $S\lt 5\pi/6$ (a point swept over twice is counted only once).