CMO 1987 Problem 4

Five points are arbitrarily put inside a given equilateral triangle $ABC$ whose area is equal to $1$ . Show that we can draw three equilateral triangles within triangle $ABC$ such that the following conditions are all satisfied: i) the five points are covered by the three equilateral triangles; ii) any side of the three equilateral triangles is parallel to a certain side of the triangle $ABC$ ; iii) the sum of the areas of the three equilateral triangles is not larger than $0.64$ .