ILL 1984 Problem 18

Let $c$ be the inscribed circle of the triangle $ABC$ , $d$ a line tangent to $c$ which does not pass through the vertices of triangle $ABC$ . Prove the existence of points $A_1,B_1, C_1$ , respectively, on the lines $BC,CA,AB$ satisfying the following two properties: $(i)$ Lines $AA_1,BB_1$ , and $CC_1$ are parallel. $(ii)$ Lines $AA_1,BB_1$ , and $CC_1$ meet $d$ respectively at points $A' ,B'$ , and $C'$ such that $$\frac{A'A_1}{A' A}=\frac{B'B_1}{B 'B}=\frac{C'C_1}{C'C}$$