The figure below shows a triangle
ABC with equilateral triangles ABC_{1}, BCA_{1},
and ACB_{1} of centroids C_{2}, A_{2},
and B_{2}, respectively. If A_{3}, B_{3},
and C_{3} are the midpoints of AA_{2}, BB_{2}, and CC_{2},
respectively, prove that the triangle A_{3}B_{3}C_{3}
is equilateral.
