ILL 1985 Problem 73

Let $A_1A_2,B_1B_2, C_1C_2$ be three equal segments on the three sides of an equilateral triangle. Prove that in the triangle formed by the lines $B_2C_1, C_2A_1,A_2B_1$ , the segments $B_2C_1, C_2A_1,A_2B_1$ are proportional to the sides in which they are contained.