1987 CMO 第 4 题

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 $0.64$ .

在给定的等边三角形 $ABC$ 内任意放置五个点，其面积等于 $1$ 。证明我们可以在三角形 $ABC$ 内绘制三个等边三角形，并且满足以下条件：

i) 这五个点被三个等边三角形覆盖；

ii) 三个等边三角形的任意一条边与三角形 $ABC$ 的某条边平行；

iii) 三个等边三角形的面积之和不大于 $0.64$ 。