SMO 1995 Problem 3

Let $P$ be a point inside $\vartriangle ABC$ . Let $D, E, F$ be the feet of the perpendiculars from $P$ to the lines $BC, CA$ and $AB$ , respectively (see Fig. ). Show that (i) $EF = AP \sin A$ , (ii) $PA+ PB + PC \ge 2(PE+ PD+ PF)$