Answer to Question #279369 in Discrete Mathematics for Jaishree

Question

Suppose 5 points are chosen at random in the interior of an equilateral triangle

T where each side has length two inches. Show that the distance between two of

the points must be less than one inch.

Answer 1

Partition the triangle into four smaller equilateral triangles by connecting the midpoints of the three sides of the big triangle.

Each of the small triangles has sidelength one inch. By the pigeonhole principle at least two of the points must be in the same small triangle, hence their distance is bounded by one inch.