NL-NWO 2006 Problem 4

Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$ . The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$ . The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$ . The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$ . Name the centers of the inscribed circles of triangle $AED$ , triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$ . Prove that $r_A + r_B + r_C = r$ .