Answer to Question #250 in Geometry for Robert Harvey

Question

Five points are located within an equilateral triangle with the sides of 1. Prove that the distance between each pair of them is less than 0.5.

Answer 1

The mid-lines of an equilateral triangle with the sides of 1 divide it into four equilateral triangles with sides of 0.5. Therefore, in one of them there are at least two given points and these points can not be situated at the top of the triangle. The distance between these points is less than 0.5.