IL-GILLIS 2014 Problem 2

Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered clockwise. It is given that $A_3=B_3=C_3$ . Let $M$ be the center of mass of $\Delta A_1B_1C_1$ , and let $N$ be the center of mass of $\Delta A_2B_2C_2$ .

Prove that $\Delta A_3MN$ is an equilateral triangle.