Question

If we select 10 points in the interior of
an equilateral triangle of side 1, show that
there must be at least 2 points whose
distance apart is less than 1/3.​

Answer 1

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Answer 2

We are given an Equilateral triangle with side 1 and we know that any equilateral triangle can be divided into 9 equal equilateral triangles and whose each side will be one third of the original one. And in total there will be 10 points.

therefore in our case, the smaller equilateral triangle will have sides 1/3 each.

Accordingly we can say that at least two of the formed ten points must lie within the smaller triangle and the distance between these two points can be at most the length of the triangle that is 1/3.