ÖMO 1988 Problem 2

An equilateral triangle $A_1 A_2 A_3$ is divided into four smaller equilateral triangles by joining the midpoints $A_4,A_5,A_6$ of its sides. Let $A_7,...,A_{15}$ be the midpoints of the sides of these smaller triangles. The $15$ points $A_1,...,A_{15}$ are colored either green or blue. Show that with any such colouring there are always three mutually equidistant points $A_i,A_j,A_k$ having the same color.