Given a triangle $ABC$ and points $D, E, F$, which are points of contact of the inscribed circle to the sides of the triangle. i) Prove that $\frac{2pr}{R} \le DE + EF + DF \le p$ ($p$ is the semiperimeter, $r$ and $R$ are respectively the radius of the inscribed and circumscribed circle of $\vartriangle ABC$). ii). Find out when equality is achieved.