ILL 1971 Problem 25

Let $ABC,AA_1A_2,BB_1B_2, CC_1C_2$ be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments $A_2B_1,B_2C_1, C_2A_1$ by $P,Q,R$ in this order. Prove that the triangle $PQR$ is equilateral.