TW-TST 1998 Problem 4

Let $I$ be the incenter of triangle $ABC$ . Lines $AI$ , $BI$ , $CI$ meet the sides of $\triangle ABC$ at $D$ , $E$ , $F$ respectively. Let $X$ , $Y$ , $Z$ be arbitrary points on segments $EF$ , $FD$ , $DE$ , respectively. Prove that $d(X, AB) + d(Y, BC) + d(Z, CA) \leq XY + YZ + ZX$ , where $d(X, \ell)$ denotes the distance from a point $X$ to a line $\ell$ .