VMO 1999 Problem 2

let a triangle ABC and A',B',C' be the midpoints of the arcs BC,CA,AB respectively of its circumcircle. A'B',A'C' meets BC at $A_1,A_2$ respectively. Pairs of point $(B_1,B_2),(C_1,C_2)$ are similarly defined. Prove that $A_1A_2 = B_1B_2 = C_1C_2$ if and only if triangle ABC is equilateral.