International Mathematical Olympiad 1996 Problem 5

Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, and $CD$ is parallel to $FA$. Let $R_A, R_C, R_E$ denote the circumradii of triangles $FAB, BCD, DEF$, respectively, and let $P$ denote the perimeter of the hexagon. Prove that

$$R_A + R_C + R_E \geq \frac{P}{2}.$$