Prove that among any five points selected inside an equilateral triangle with side equal to 1, there always exists a pair at the distance not greater than .5.

Proof

Prove that among any five points selected inside an equilateral triangle with side equal to 1, there always exists a pair at the distance not greater than .5.

## Proof

Indeed, split the triangle into four smaller ones by connecting midpoints of its sides. The largest possible distance between two points of one small triangle is given by the length of its side which is .5. Now, we are given 4 triangles and 5 points. By the Pigeonhole Principle, at least one triangle contains at least two points. The distance between any two such points does not exceed .5.

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