Five points are located in an equilateral triangle with 10-inch sides (or on its perimeter).
What’s the maximum distance between the two closest points?
(In reply to A guess
by Steve Herman)
Divide the large triangle into four smaller equilateral triangles by joining the midpoints of the large triangle's sides. Each small triangle will have sides of 5 inches.
Apply the pigeon hole principal with five points distributed among the four small triangles. At least one of the triangles must have two of the points in it. The distance between those two points cannot be less than the smallest distance between any two points.
Within a 5 inch equilateral triangle the furthest apart two points can be is 5 inches. The maximum distance between the two closest of the five points cannot be more than this distance. Steve's arrangement shows that a 5 inch solution is indeed achievable.