VN-TST 2004 Problem 5

Let us consider a convex hexagon ABCDEF. Let $A_1, B_1,C_1, D_1, E_1, F_1$ be midpoints of the sides $AB, BC, CD, DE, EF,FA$ respectively. Denote by $p$ and $p_1$ , respectively, the perimeter of the hexagon $A B C D E F$ and hexagon $A_1B_1C_1D_1E_1F_1$ . Suppose that all inner angles of hexagon $A_1B_1C_1D_1E_1F_1$ are equal. Prove that $$p \geq \frac{2 \cdot \sqrt{3}}{3} \cdot p_1 .$$ When does equality hold ?