OBM 2018 Problem 1

We say that a polygon $P$ is inscribed in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$ . We also say in this case that $Q$ is circumscribed to $P$ . Given a triangle $T$ , let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$ . Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs.