Pigeonhole principle, choosing point in a region

Question

Consider the following region: It is bounded by a regular hexagon whose sides are of length 1 unit. Show that if any 7 points are chosen in this region (hexagon), then 2 of them must be no further apart than 1 unit.

If I draw a line segment from the centre to each vertex how to prove that 2 points MUST BE NO FURTHER APART THAN 1 UNIT length..

The edit completely invalidated the answer already given. If you have a new question, use the "Ask Question" link in the top menu. Editing is for improving existing questions. — Asaf Karagila, May 23 '15 at 06:44
See also: Given seven points inside a hexagon with side length $1$ prove that there exists two points with distance at most $1$. — Martin Sleziak, Jul 01 '20 at 14:30

PSPACEhard · Answer 1 · 2015-05-15T09:50:52.843

3

From the center of the hexagon, draw a line to each vertex. This will partition the hexagon into 6 equilateral triangles, each with side of length 1. If 7 points are chosen, then there must be 2 points being in a same triangle according to pigeonhole principle and the distance between these two points should not be greater than 1. Please see the figure below.

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edited May 15 '15 at 09:50

answered May 15 '15 at 09:40

PSPACEhard

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Pigeonhole principle, choosing point in a region

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