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Consider six distinct points in a plane. Let $m$ and $M$ denote respectively the minimum and the maximum distance between any pair of points.

Show that $M/m \geqslant \sqrt{3}$.

Captain Lama
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Sourasekhar Banerjee
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1 Answers1

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Assume otherwise. Pick two points $A,B$ at distance $M$ (The small dots in the following image)

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Then all other points are in the closed lens-shaped area bounded by the circles of radius $M$ around $A$ and $B$ (big circles in above image). In particular, they must be in the ten blue triangles shown. As the blue triangles have side length $\frac M{\sqrt 3}$, each of them can contain at most one of the points (including on the boundary). After taking away those already occupied by $A$ and B$, there are only four blue triangles left. We conclude that each of the top two and each of the bottom two blue triangles contains one of the four remaining points.

Next consider the red lines: The shape bounded from above by the top red line and from below by the two small circles has diameter $<\frac M{\sqrt 3}$, hence can contain at most one point. We conclude that one of the points must be above the top red line. By the same argument, one of the points must be below the bottom red line. But then these two points are more than $M$ apart - contradiction!

Hagen von Eitzen
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  • How do you derive the contradiction? It is easy to find two points in one such area which are between $\frac{M}{\sqrt3}$ and $M$ apart. – almagest Apr 16 '16 at 13:42
  • A more detailed sketch will make that clearer - coming soon – Hagen von Eitzen Apr 16 '16 at 13:42
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    Sorry, but this proof CAN'T POSSIBLY be correct. It would work with just five points, and for five points the result is not true. (Consider a regular pentagon). https://www.wolframalpha.com/input/?i=diagonal+of+regular+pentagon Congrats on the upvotes! –  Apr 16 '16 at 14:02
  • @mathguy OK, a bit more of clarification was needed. But with the right sketch and additional arguments, it should now be clearer – Hagen von Eitzen Apr 16 '16 at 14:20
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    @Hagen: Don't be dishonest. This is not a "clarification" but replacing your incorrect argument with a correct one. You stated something quite different in your first posting; editing it is fine, being dishonest is not. –  Apr 16 '16 at 14:29